It was the Swansea-based region's third successive win following a nightmare start to the season, in which they lost their three opening encounters.
They temporarily climb to fifth in the league table but can be overtaken by Cardiff Blues or Leinster, who are both in action on Saturday.
Biggar missed his first shot at goal but made amends to hand the Ospreys the lead on six minutes.
Gonzalo Garcia hit-back for the hosts but Biggar kept the scoreboard ticking and the Italian minnows were dealt a blow when scrumhalf Alberto Chillon was yellow carded for a cynical foul.
The Ospreys laid siege in Zebre territory and eventually broke the defence for Fussell to grab his second try in as many games.
The in-form fly-half was again on target and the Ospreys established a commanding lead when Kahn Fotuali'I fired a long pass to send Thomas over.
Daniel Halangahu chipped away at the Ospreys' lead with another penalty to give his side a glimmer of hope, trailing by 21-6 at the interval.
Zebre continued their mini revival with a try from Luca Redolfini, who crashed through the visitors defence, with Halangahu converting.
But Biggar continued to put the boot to the Italians who were also dealt a blow when Matteo Pratichetti was shown yellow.
He then crossed the whitewash in the closing stages and converted to put the result beyond doubt.