Mediaeval compilations referencing older, potentially ancient works do mention the story.
It is narrated in Ch. II of the "Διηγηματα" section of the "Appendix: Narrationum," on p. 359 of Anton Westermann's 1843 book Mythographoi: Scriptores poeticae historiae graeci.
Cited as the source here is the Progymnasmata 2, by Aphthonius of Antioch (c. 4th century AD), as well as a commentary on the same work by John Doxopatres (11th century AD).
The story also occurs in the Geoponika [Geoponica], "Agricultural Pursuits," which Wikipedia describes as "a twenty-book collection of agricultural lore, compiled during the 10th century in Constantinople for the Byzantine emperor Constantine VII Porphyrogenitus."
According to the relevant passage thereof, i.e. Book 11, Chapter 17, entitled "Concerning the Rose",
Let him that admires the beauty of the rose, reflect on the wound of
Aphrodite, they say; for the goddess indeed loved Adonis, and Ares on
the other hand loved her: but Ares in a fit of jealousy killed Adonis,
thinking that the death of Adonis would put an end to her affection
for him; but the goddess, having understood what had been done,
hastened to be revenged; and throwing herself in a hurry on the rose,
when without her sandals, she was wounded by the thorns of the rose in
the sole of her foot; and the rose, which was before white, from the
blood of Aphrodite, changed into the colour in which it is now seen,
and it became red and sweet-scented. But others say that, when the
gods were feasting above, and there stood a great quantity of nectar,
Eros led the dance, and with his wing struck the bottom of the bowl
and overturned it, and that the nectar poured on the ground made the
rose of a red colour.
On p. 226 (Footnote 3) of Vol. 1 of his 1914 book The Golden Bough, James George Frazer cites John Tzetzes' commentary on Lycophron's Alexandra 831 as a source for the myth, but I have struggled to find any explicit mention of such a thing therein, beyond a reference to the Muses weeping for Adonis upon his death.
Alexandra is an extremely cryptic poem narrated essentially in riddles, though, so it might just be that I don't get it, even with Tzetzes' commentary. If Lycophron did indeed allude to the story in Alexandra, this would be an undoubtedly ancient primary source for it, several centuries older than the aforementioned compilations.
Also ancient but sometime later than Lycophron, Bion's Epitaph of Adonis (commonly referred to in English also as Lament for Adonis) tells a slightly different version of the story, in which Aphrodite cuts herself on thorns as she rushes through the woods on her way to the fallen Adonis, but in the end it is Adonis' blood which turns into roses. Aphrodite in turn sheds a tear for each drop of his blood, and these tears turn into windflowers (anemones).