Their Russian opponents were a match for the Italian side in the first-half which ended with the scores locked at 14-14.
But the visitors slowly took charge after the interval with fly-half Carlo Canna taking his tally to four penalties and a conversion.
Scrum-half Guglielmo Palazzani added his second try of the game and four minutes from time South African forward replacement Johan Meyer added Zebra’s fourth try to round off the scoring.
Palazzani had opened the scoring with his first try after just three minutes and Canna also crossed before the break.
Enisei-STM battled back with tries from full-back Nikita Churashov and flanker Mikheil Gachechiladze both converted by fly-half Yuri Kushnarev to level the scores before the break.
The sides clash again in Parma next Saturday with Enisei-STM looking to record their first win in Pool 4.