The flanker was in imperious form as the Swansea-based region got back to winning ways following three defeats on the bounce.
And a penalty try and a touchdown from Scott Baldwin ensured the Ospreys claimed a bonus-point.
Tom Prydie and Francisco Chaparro crossed for the Dragons but their woes were compounded by the injury to Faletau.
The Wales number eight received 10 minutes of treatment on the pitch following a heavy collision and was taken to hospital.
“He had some pain in his neck and with anyone with a neck injury, particularly on the rugby field, you’ve got to take extra precautions,” said Dragons medical officer Dr Rohit Kulkarni.
“Our suspicion is there’s not much happening but we’ve got to be very careful, with neck injuries in particular.
“I would suspect by the next day or so we’ll have a clearer picture as to what’s happening with Toby.”
The Dragons made a perfect start to the Liberty Stadium encounter with Prydie crossing after just four minutes against his former region.
Kris Burton added the extras but the Ospreys quickly exerted their dominance.
Biggar chipped away at the Dragons’ lead and a dazzling break from Matthew Morgan put Tipuric over for his first.
A 10-point purple patch early in the second-half including Tipuric’s second put the result beyond doubt.
The Dragons were dealt a blow when Ieuan Jones was yellow carded within minutes of coming on and the Ospreys took full advantage by forcing a penalty try at a scrum.
Scott Baldwin crossed for the bonus-point with ten minutes to spare before Chaparro grabbed a late consolation.Â
