The Ospreys trailed the Italian minnows until the 66th minute in an error-strewn encounter at the Liberty Stadium.
But Biggar slotted three penalties and scored a converted try to overcome the PRO12 basement boys, who scored tries through Alberto Chillon and Tito Tebaldi.
The Ospreys were below par from the kick-off, with several uncharacteristic mistakes handing the ascendancy to the Italians.
Chillon opened the scoring for the visitors and he stretched their lead with the game’s opening try, from close-range, on 28 minutes.
Biggar quickly hit-back with a pair of penalties and Zebre were dealt a further blow when Andries van Schalkwyk was sent to the sin-bin.
Both Biggar and Daniel Halangahu missed shots at goal in the second-half but the game was sparked into life with the Ospreys fly-half slotting a penalty and grabbing a superb try.
Tom Isaacs launched the attack with a scintillating break and, after the ball was spread through the hands, Biggar crossed the whitewash.
Zebre almost instantly responded with their second try to set-up a tense finale but the Ospreys hung on to claim the slender triumph.
