The Ospreys had probably targeted a bonus point win against the winless Italians but an error-strewn display saw them trail until the 66th minute.
Biggar scored three penalties and converted his own try, while Zebre scored tries through scrum-half Alberto Chillon and his replacement Tito Tebaldi.
It was the Italians who got on the scoreboard first through a penalty goal for Chillon following a ruck offence and Ospreys trailed by eight after a couple of gaps opened up allowing Chillon to dive over for the first try of the game after 28 minutes.
The Ospreys closed the gap to two points five minutes later with two Biggar penalties before Zebre number eight Andries van Schalkwyk was yellow carded as the home side drove for the line.
Despite leading 8-6 Zebre made five changes at the interval including the entire front row and try scorer Chillon.
The Ospreys, who brought on in-form scrum-half Kahn Fotuali'i after 54 minutes to loud cheers, continued to show frailties in defence and it was only a knock-on in their 22 which prevented the Italians from claiming a second try.
On the hour Zebre failed to take the chance to go five points up when replacement centre Daniel Halangahu was wide with a penalty shot.
The Ospreys suddenly came to life with Biggar putting over a third penalty as they led for the first time before he went over for a converted try as the home side put the ball impressively through the hands.
But the Welsh side's joy was short-lived as Zebre hit back with a second try three minutes later when Tebaldi slid in at the left corner to cut the Ospreys' lead to only a point.
The hosts held on but it was not the sort of performance coach Steve Tandy would have wanted before they host Leicester in the Heineken Cup next Sunday.
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