But flanker Tom Smith crossed shortly after the interval and with Biggar's boot on form the Ospreys held on for their third successive victory.
With just a point to show from their opening two games it was Treviso who settled into the game the quicker as they tried to turn their woes around, Kris Burton kicking a fourth minute penalty.
But after 18 minutes the sides were locked at 6-6, Ospreys fly-half Biggar matching Burton kick for kick.
However, the home fans had plenty to shout about on 23 minutes, centre Morisi going over, Burton failing to add the extras for a five point lead.
A yellow card for No.8 Marco Filippucci let Ospreys back into the game as Biggar was successful with another penalty on 35 minutes, only for Treviso to pull away moments later thanks to Vermaak's converted try.
With the clock ticking down the visitors looked to have it all to do in the second half only for scrum-half Tom Isaacs to touch down in his first start in the RaboDirect Pro12 from Andrew Bishop's crossfield kick.
Biggar kicked the conversion to close the gap by just two points at the interval.
And Ospreys were clearly buoyed by this, flanker Smith crossing two minutes after the restart, Biggar again with the extras.
This was the first time Ospreys had led the Italians and Biggar was in no mood to let it go, kicking a penalty and a drop goal to extend their lead to 11 points.
Replacement Willem de Waal did his best to turn the tables late on with three penalties, the final two with Biggar in the sin bin, but Ospreys' Kristian Phillips knocked one over of his own with a minute left to put the game beyond doubt.
|Benetton Treviso Score Card|
|Willem de Waal||3||9|
|Ospreys Score Card|