However, the Welsh region, having lost six of their last seven matches in all competitions, were forced to come from behind in the first half at the Liberty Stadium.
The Ospreys were indebted to second-half tries from flanker Tyler Aldron and veteran prop Duncan Jones - his first score for four years.
Zebre, who had beaten the Blues 30-25 and drawn with the Scarlets already this season, scored tries through flanker Filippo Cristiano and number eight Samuela Vunisa.
The Ospreys had been boosted by the return of wing Hanno Dirksen, making his first Pro12 start since undergoing knee surgery in November 2012.
Early on, Zebre showed their love of playing Welsh sides by scoring the only try of the first half, although they trailed at the interval 15-13 to five penalties from Morgan.
It was Morgan who gave the region a confident start with two penalties after two and 11 minutes, and the Ospreys could also have had a try had flanker Aldron not lost the ball over the line.
It was Zebre who claimed the first try, however, when Cristiano finished off a 60-yard run from back-row colleague Andries van Schalkwyk, who picked off a pass from Morgan.
After that the half became a kicking contest, with Morgan finding the target three more times, the third to ensure the Ospreys went in ahead at half-time.
Zebre also had penalties from outside-half Paulo Buso and captain Gonzalo Garcia.
After half-time the Ospreys went a long way to sealing their win with two tries in four minutes.
The first was a well-worked move started by some good link work between full-back Richard Fussell and Dirksen before scrum-half Tito Tebaldi's long pass put Aldron over.
That was followed up by Jones' try - only his fourth in 196 Ospreys appearances - after some good build-up work by centre Ben John.
Despite trailing 27-13, Zebre failed to throw in the towel and responded with a try for Vunisa. However, it was his sin-binning for a professional foul that probably put paid to any further points for the Italians.
Five minutes from time Morgan then kicked his sixth penalty to seal the victory.
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- Duncan Jones