Early Medieval Chinese History Ph.D. student here.

The section on the Western Regions (a term for areas west of China proper, modern Xinjiang and on) in the Book of Later Han records the first mission as being in 166 CE, during the reign of Emperor Huan of the Han Dynasty. Rome is referred to as Daqin 大秦, though the description of Rome is somewhat fanciful.

The mission is recorded as such (quick translation by me):

至桓帝延熹九年，大秦王安敦遣使自日南徼外獻象牙、犀角、玳瑁，始乃一通焉。

"In the ninth year of Yanxi in the reign of Emperor Huan, the king of Daqin Andun dispatched emissaries that came through Rinan, outside the frontier, to present elephant tusks, rhinoceros horns, and tortoise shells. [This is] when communications first began."

Several things are noteworthy:

1. Andun is probably Marcus Aurelius Antoninus.
2. The victory over Parthia probably allowed Romans more easy access to trading routes on the Indian Ocean; the emissaries almost certainly came up through Southeast Asia, as the place they met the Han was in present-day Vietnam
3. The goods they brought are probably not the goods they intended to bring to present. None of those things are especially "Roman" in nature, and are in fact all things easily obtained in Southeast Asia. My adviser theorizes that they probably got shipwrecked on the way to China and had to buy local products instead.

AFAIK there is no Roman record of this particular mission.