You Crush the Yuuzhan Vong

[Sionnach Glic]

Well, one species could do it, if they had overwhelming numbers. (cough) Tyranids (/cough)

The Tyranids are the concept of an extra-galactic alien biotech race done right. They're not particularly uber one on one, they win purely due to massively overwhelming numbers, throwing thousands of Gaunts at the enemy purely to wear out their ammunition.
Hell, a fleet of 1.6 billion of their ships (yes, billion with a b) was considered small.

[Tsukiyumi]

I have a bio-tech race in my sci-fi universe, but they

A) also use "standard" tech, like FTL drives, some weapons, shielding and sensors (basically, their ships are individual organisms grown around manufactured tech), and

B) use massive numbers in place of more capable technology.

The idea of a pure bio-tech race that's even close to being on par with a galaxy-spanning civilization seems far-fetched to the extreme IMO.

[Deepcrush]

That and how pathetic numbers the YV numbers were. 11,000 cap ships...