Tito Tebaldi's try, to add to the score by the scrum-half he replaced, Alberto Chillon, gave the basement boys hope but try as they might, they could not edge back in front and Ospreys escaped with the four points.
Ospreys didn't have the bests of starts, even before kick off they were dealt a blow when Andrew Bishop was ruled out, meaning Jonathan Spratt was thrown into the starting XV.
And Zebre capitalised on the Ospreys' sluggish start, drawing first blood on 12 minutes when the hosts were penalised for not rolling away - visiting scrum-half Chillon stepping up to kick the three points.
That breathed some life into the Ospreys but after a rare break through the Zebre defence, Rhys Webb lost his footing and the chance was gone.
Biggar, fresh from booting Ospreys to a commanding victory over Scarlets last time out, drew his side level on 22 minutes after Zebre were penalised for not rolling away and he made no mistake from 45 metres.
Zebre were undeterred however and the Ospreys try-line was beckoning when the hosts intercepted but that only handed the defending RaboDirect PRO12 champions a stay of execution.
On 28 minutes a fine push from the Zebre forwards got Chillon over the line to restore the bottom club's lead, although he could not add the extras and Biggar trimmed the gap to just two after some strong forward play from the Ospreys.
Desperate to take the lead before half-time, Ospreys raised their game and began to knock on the Zebre door, prompting visiting No.8 Dries van Schalkwyk to come in at the side and receive a yellow card for his troubles on 39 minutes.
But as the Ospreys forwards attempted to force their way over, they were held up over the line and as a result, Zebre took an 8-6 lead into the break.
Despite their lead, Zebre had struggled in the scrum in the first half and took the drastic action of changing their whole front row, bringing on Carlo Festuccia, Carlo Fazzari and Luciano Leibson.
But in the first scrum of the second half, the Ospreys earned another penalty, only for Biggar to miss the target from 35 metres out.
With 23 minutes remaining, Zebre were then given the chance to extend their lead when full-back Matthew Morgan was penalised but replacement Daniel Halangahu missed his kick from 40 metres.
And it came back to haunt the visitors when Zebre were pinged for being offside with 64 minutes gone, this time Biggar hitting the target to give his side the lead for the first time in the match.
That seemed to knock the wind out of Zebre's sails and just two minutes later, Tom Isaacs made the break for the Ospreys before handing over to Jones who in turn found the supporting Biggar to touch down.
He converted his own try but no sooner had the Ospreys faithful breathed a sigh of relief than Zebre scrum-half replacement Tebaldi nipped over for a try, converted by Halangahu, to ensure the home side led by just a point at 16-15 with eight minutes to go.
That galvanised Zebre who pushed all the way for their first ever RaboDirect PRO12 win and were camped on the Ospreys try-line but they couldn't find a way through and had to settle for a losing bonus point.
|Ospreys Score Card|
|Zebre Score Card|