Second-half tries from Martyn Thomas and Lewis Robling gave the Dragons the initiative with fly-half Jason Tovey kicking 11 points in a full-blooded encounter.
The hosts had their noses in front for most of the match but four penalties from Matthew Morgan kept the Ospreys in touch before Tommy Bowe's try set up a grandstand finish in which Biggar twice missed a chance to win the bragging rights with an off target conversion and drop goal.
Ospreys head coach Sean Holley made ten changes to the side that downed Cardiff on New Year's Day while for Dragons hooker Lloyd Burns pulled out late with illness and was replaced by Steve Jones.
The Dragons, captained by blind side flanker Danny Lydiate for the first time, took the lead in the ninth minute through Tovey, who made no mistake with an early penalty.
The Ospreys cause went from bad to worse when Ian Evans was sin-binned for coming over the top, but Tovey failed to increase the Dragons' advantage from close-range on 12 minutes - but the Wales U20 international made amends two minutes later from a similar position.
Opposing fly-half Morgan, who drifted a sixth minute penalty wide, then restored parity for the Ospreys with kicks in the 25th and 34thminute in what was proving to be a cagey first half at Rodney Parade.
In a pulsating end to the half though, both sides were reduced to 14 men when firstly Dragons winger Tonderai Chavhanga and then Ospreys' Tom Smith were sent to the sin bin, before Tovey sent over his third penalty of the half to give the hosts a 9-6 lead at the break.
Not to be outdone, Morgan fired over his third penalty five minutes after the restart but almost immediately Dragons' full-back Thomas grabbed the game's first try, going over in the right corner after a half-break from Tovey exposed the space in the Ospreys defence.
Tovey pulled the conversion wide, but two minutes later a poor Morgan pass presented the ball straight to Dragons substitute Robling, who ran unchallenged from 40 metres to touch down underneath the posts. Tovey added the extras for a 21-9 lead.
With the game slipping away, Ospreys coach Scott Johnson brought some of his big guns off the bench, with Huw Bennett, Duncan Jones and Rhys Webb all joining the fray.
It took immediate effect, with Morgan sending over a long-range penalty after good work by the forwards, with his replacement Biggar adding another three points just after the hour mark to bring the Ospreys within one score.
They got that score, as Biggar, whose pass looked suspiciously forward, fed Bowe to power over seven minutes from time but the outside half missed the conversion before his dramatic drop goal attempt came back off the post.
|Newport Gwent Dragons Score Card|
|Ospreys Score Card|