The Swansea-based region have taken many by surprise with their early-season form, but came close to losing their unblemished record in Parma.
Fly-half Dan Biggar's early exit following a collision with Leonardo Sarto came as a huge blow to the visitors and placed considerable pressure on replacement number 10 Sam Davies to deliver with the boot.
And, despite missing four efforts from the tee, the former Wales Under-20 international accounted for all 15 of his side's points.
Giulio Bisegni scored the only try of the match in the fourth minute, with the counter-attacking move sparking hopes of a second shock result in succession after Zebre's 13-6 win over Ulster last weekend.
For the Ospreys, meanwhile, the touchdown will have evoked unwelcome memories of their 30-27 defeat at Stadio XXV Aprile in May.
Giulio Toniolatti was sin-binned for killing possession illegally 16 minutes into the second half, before - in a game that was all about fine margins - Gonzalo Garcia hit the crossbar with a penalty from his own half.
Davies swiftly punished the hosts by kicking his side in front for the first time and the youngster, who turns 21 on Monday, hauled Ospreys back in front once again nine minutes from time after Garcia had restored Zebre's advantage.
Ospreys have nevertheless lost their place at the top of the Pro12 standings to a Glasgow Warriors outfit who have maintained their momentum from last season.
The 2013-14 runners-up are also unbeaten this term, but ran in six tries on their way to claiming a 40-23 triumph at Benetton Treviso on Sunday.
Tommy Seymour touched down twice in the opening 40 minutes for the visitors, but it was James Eddie who got the all-important fourth try to earn a bonus point that sends Glasgow above Ospreys at the summit.
Champions Leinster continue to struggle at the start of the new campaign and were left to rue a poor first-half display in a 34-23 defeat to Munster at the Aviva Stadium.
Matt O'Connor's men conceded three tries to go in 28-9 down at half-time and failed to take advantage of their visitors' four second-half sin-binnings, although they did manage to make a small dent in the deficit through a penalty try and a Darragh Fanning touchdown.
Elsewhere, Ulster consigned Edinburgh to a 30-0 defeat, Cardiff Blues staged a late fightback to snatch a 24-24 draw at Connacht and Scarlets beat Newport Gwent Dragons 26-13.