Winger Ben John and scrum-half Rhys Webb also scored tries, with Biggar converting twice - from his own score and that of Webb's which came from the final play of the match.
The victory keeps Ospreys in contention to retain their RaboDirect PRO12 title, though they missed out on a bonus point that could yet prove crucial.
Steve Tandy's men had fallen out of the top four ahead of kick off after fellow Welsh region Scarlets beat Glasgow last night to leapfrog them in the play-offs.
And the hosts' plans were dealt a further blow as they were forced into a late change at the Liberty Stadium when Kahn Fotuali'i, who is set to leave the club for Premiership side Northampton Saints in the summer, was forced to withdraw from the team due to illness, Webb taking his place at scrum-half.
It was the visitors who struck first as their scrum-half Tobias Botes kicked a seventh-minute penalty following a tight opening to the match.
But the Italians were unable to kick on from that early advantage, and their lead was soon cancelled out with an Ospreys penalty, Biggar pulling his side back level at 3-3, after striking an earlier attempt against the post.
The Ospreys fly-half looked in good form though, and it was his boot that soon had the hosts in control and by the 32nd minute he had kicked two further penalties following poor defence from the visitors for a 9-3 lead.
They had their first try of the match a minute before half-time, and it was gift-wrapped to them by the Italians who gave possession away for John to crash over and increase his side's lead to 14-3, Biggar blotting his copybook by missing the conversion.
In the second half, Ospreys immediately set about the Treviso defence in a bid to increase their lead, but they found the visitors in resolute mood, clearly showing defiance buoyed by recent victories over Cardiff Blues and Munster, and a draw in Belfast against Ulster in their previous three outings.
Eventually though the pressure told and Biggar again was at the centre of it as he picked up a loose ball after poor handls from Treviso and slid over the try line ten minutes after the break, before converting for a 21-3 lead.
For much of the second half there were few attacking opportunities for either side, though Ospreys continued to force the issue knowing a bonus point could prove vital in their ongoing battle with Welsh rivals Scarlets.
But despite enjoying a man advantage for ten minutes from the hour mark when Bees Roux was sent to the sin bin for Treviso, it wasn't until the final minute of the match that they could increase their lead as Webb finished off a good move, Biggar adding the extras.
|Ospreys Score Card|
|Benetton Treviso Score Card|