Munster clinched a second straight Guinness PRO14 bonus-point win but were made to work for it on Sunday afternoon in Parma.
The province remain second in Conference A, 12 points behind Glasgow.
Tries from Rory Scannell (18) and James Hart (25) - the latter his first for the province - were both converted by JJ Hanrahan, who had opened the scoring with a 10th-minute penalty, as the visitors stormed out to a 17-0 lead with less than half an hour on the clock.
Munster were reduced to 13 men following yellow cards for Robin Copeland and Sam Arnold, and Zebre took advantage to get back within touching distance before half-time.
Giulio Bisegni (36) and captain Jacopo Sarto (40) touched down and Guglielmo Palazzani added the extras to pull the Italian side back to 17-14.
Alex Wootton scored further tries on 52 and 56 minutes, Hanrahan converting the second, to get Munster clear once more and, although Giovanbattista Venditti's try 11 minutes from time again gave Zebre a glimmer of hope, that was quashed when Calvin Nash got in on the act.
The 20-year-old Nash - set up by fellow academy player Jack Stafford - crossed for his first Munster try and Hanrahan made no mistake from the tee to seal a comfortable away victory.