In fact there are at least two such stories which one might easily call typical happily-ever-after fairy-tale-style narratives, one of which is quite well-known and the other not enjoying quite as much renown.
The more famous story is that of Pygmalion and Galatea, though this Galatea is different from the Sicilian Nereid who gets involved with Acis and Polyphemus.
Epaphus' daughter, Princess Lysianassa of Egypt, bore the sea-god Poseidon a son, Pygmalion, who became king of the island of Cyprus, where he carved from ivory an image of the goddess Aphrodite. He fell in love with this artwork of his, which he would kiss, caress, embrace, clothe and adorn with jewels and other finery which he offered to it as gifts.
At the festival of Aphrodite he prayed to the gods to give him a wife who was like his ivory statue, and Aphrodite gave life to the sculpture itself so that it became a real woman. Pygmalion's (formerly sculptural) wife shortly afterwards bore him a daughter called Paphos, after whom the famous Cyprian city of Paphos was named. Aphrodite's sometimes-tragic lover Adonis was the grandson of Paphos.
The Wikipedia article on Galatea notes that:
Though the name "Galatea" has become so firmly associated with
Pygmalion's statue as to seem antique, its use in connection with
Pygmalion originated with a post-classical writer. No extant ancient
text mentions the statue's name... According to Meyer Reinhold, the
name "Galatea" was first given wide circulation in Jean-Jacques
Rousseau's scène lyrique of 1762, Pygmalion.
The other story is that of Daphnis and Pimplea. The demigod poet herdsman Daphnis, a son of Hermes, is typically a tragic love-story character in the works of better-known writers such as Parthenius, Theocritus, Aelian, Virgil and Diodorus Siculus. There is a fragment, however, which remains of an otherwise lost play by Sositheus, which tells of how Daphnis became king of Phrygia.
King Lytierses of Phrygia, whenever a stranger arrived in his domain, would challenge him to a contest of harvesting wheat to see who could gather the most sheaves in one day, both competitors beginning at the same time. If his opponent’s strength flagged in the course of performing the task, Lytierses whipped him. The king would always win and then behead his opponent and conceal the body in one of the sheaves while chanting a lugubrious dirge.
Daphnis, a reclusive shepherd who lived in Sicily (like Polyphemus, Acis and Galatea), tended his flocks all year long on Mt Aetna. Daphnis's sweetheart, a young woman named Pimplea, was kidnapped by pirates and sold into slavery. Daphnis searched for her "throughout the world", as Robert Graves puts it, before finding her among the slave-girls of Lytierses.
Daphnis was roped into the Phrygian king's customary contest, but it just so happened that the great hero Heracles had also presently arrived in the country, and offered to take Daphnis’ place in the competition. Heracles defeated Lytierses and decapitated him with a sickle, casting the king’s beheaded corpse into the local river Maeander. From then on Phrygian reapers honoured Lytierses with a dirge during the harvest. Heracles then enthroned Daphnis as Lytierses' successor, and for the wedding of Daphnis and Pimplea, he provided the bride with the former king's palace for her dowry.
There are several other examples too.
Lynceus and Hypermnestra
Pygmalion's half-brother and cousin Belus (for Poseidon was his father but Lysianassa's sister Libya was his mother) had twin sons Aegyptus and Danaus to whom, from his vast empire, he gave the kingdoms of Arabia and Libya as their respective inheritances. Aegyptus further conquered the territory which lay in between his own and that of his brother in Africa and named his new kingdom after himself Aegyptus, "Egypt." The twins quarrelled about this and Danaus, feeling threatened by his brother, took his fifty daughters with him to their ancestral homeland of Argos in Greece, where Danaus was awarded the throne of that city.
Aegyptus, who had wanted his fifty sons to marry his fifty nieces, sent the sons after their uncle and cousins. They caught up with them, whereupon Danaus agreed to the mass wedding but instructed the brides to kill their bridegrooms in their beds on the wedding night. Daggers having been provided for this, all the brides obeyed their father except for one, Hypermnestra (whose name means "Super-Bride"), who spared her groom, Lynceus.
For her disobedience, Hypermnestra was judged guilty by Danaus and sentenced to be cast into a dungeon, but eventually they were reconciled and Hypermnestra was reunited to Lynceus. In order to acquire new grooms for his other forty-nine children, who were now notorious as husband-slayers, Danaus held an athletic contest whose prizes were these young women, and thus they were all remarried.
Although Athena and Hermes purified Hypermnestra's sisters of their husbands' murder, after their own deaths, they were sent to Tartarus for punishment. In revenge for the death of his brothers, Lynceus later killed Danaus, whom he then succeeded as king of Argos. Hypermnestra bore Lynceus a son, Abas. Perseus, the Gorgon-slayer, was a great-grandson of Abas.
Cadmus and Harmonia
Belus himself had a twin brother, Agenor, whose son Cadmus arrived in Greece searching for his sister Europa, who had been abducted by Zeus. Cadmus founded the city of Thebes, and shortly before he built the city, he killed a dragon which was the offspring of Ares. With help from the other gods, however, he was able to marry Ares' daughter Harmonia.
The couple suffered a string of family catastrophes mostly involving their children and grandchildren but afterwards they migrated to Illyria in what is now Croatia and Albania, and battled the locals there where they founded a new kingdom. Eventually they were transformed by the gods into dragons and afterwards, having been restored to humanoid form, they were granted immortality and transferred to the Islands of the Blessed.
Dionysus and Ariadne
The wine-god Dionysus was a grandson of Cadmus and Harmonia. Princess Ariadne of Crete Island was a granddaughter of Zeus and Europa, making her Dionysus' second cousin. After Theseus abandoned Ariadne on the island of Naxos, Dionysus found her there and married her, granting her immortality.
Bïas and Pero
The brothers Melampus and Bïas were grandsons of Abas. Bïas fell in love with his own cousin Pero, and Melampus went through a convoluted adventure in order to win Pero's hand in marriage on his brother's behalf. He eventually succeeded and Bïas was wed to Pero. The Argonauts Talaus, Areius and Laodocus were their sons.
Odysseus and Penelope
Odysseus helped Tyndareus to solve the issue of the numerous suitors of Tyndareus' stepdaughter Helen. In return Tyndareus convinced his brother Icarius to give his own daughter Penelope's hand in marriage to Odysseus.
As Odysseus was taking Penelope, Icarius pursued her, imploring his daughter to stay with him. Odysseus gave Penelope the option to do just that. In response, she modestly covered her face with a veil, indicating her desire to leave with her bridegroom. On the spot where this happened, Icarius later put up a statue of Aidos, the personification of modesty.
At the core of the saga of the Odyssey is the story of Odysseus, in the aftermath of the Trojan War, trying to get back home to his wife and his son. It ends after he has succeeded, reunited with his family, whom he has not seen in twenty years. Later on Odysseus and Penelope die in peaceful old age.
(There are certain other sequels to the Odyssey, however, in which Odysseus' adventures continue. In one of these stories, Odysseus is slain by Telegonus, his own son by the immortal witch Circe, Telegonus not knowing that Odysseus is his father. Telegonus bitterly regrets the death he has caused but afterwards he even marries his father's widow Penelope, who bears him a son, Italus, who becomes a king in Italy, which gets its name from him.)
Acontius and Cydippe
Acontius was a handsome young man from the island of Ceos. At the annual festival of Artemis on the island of Delos, he fell in love with an Athenian aristocrat named Cydippe. While she was sitting in the temple, he threw before her an apple upon which he had written, "I swear by the sanctuary of Artemis to marry Acontius." Cydippe read aloud what was written thereon and then threw the apple away.
When Cydippe's father was about to give her in marriage to another man, she fell sick three times just before the nuptial solemnities were about to begin. The Delphic oracle declared that Artemis was punishing Cydippe for perjury, whereupon Cydippe explained the whole affair to her mother, and the father was induced to give his daughter to Acontius.
There is a tragic doublet of this story which Antoninus Liberalis tells in his Metamorphoses, where the young man, Hermochares, is the Athenian while the young woman, Ctesylla, is from Ceos. The narrative ends after Ctesylla has died in labour. For further detail on that, see the story of Hermochares and Ctesylla.