With Stockdale’s score, coming after the winger won the race to Stuart McCloskey’s kick ahead, having represented the last action of the first half, Benetton were quickly out of the blocks in the second.
Only five minutes had elapsed since the restart when, after Ulster were too careless in possession, Andy Uren scooped up a loose ball and sent Umaga over in the corner.
Ulster responded quickly, though. Five minutes later, with their backline completing a series of accurate, wide passes, Will Addison was in for the crucial, bonus-point try.
The win, though, still needed to be secured. 28-13 ahead after Cooney added the extras, two converted tries in the space of five minutes brought Benetton back within a point with just over a quarter of an hour remaining.
While the first of that quickfire pair, scored by Toa Halafihi, was the product of some neat footwork from Menoncello, the second again came from Ulster errors.
After Mike Lowry lost his footing chasing a kick in behind, Ulster were stripped on their own line to give Ivan Nemer the simplest of scores.
Benetton made a mistake of their own on the restart, though, and after a few phases hammering at the Benetton line, Cormac Izuchukwu was rewarded with his first senior try.
Again Cooney converted and his 73rd minute penalty rounded off Ulster’s scoring, although the late score from Tomas Albornoz ensured his side would bring two points back to Italy.