They got off to a perfect start when Nick Williams' first of the night moved them into a 13-0 lead with barely a quarter of an hour gone but from there they had to work hard for their scores.
Williams added another on the half hour mark with lock Peter Browne providing the third just as the contest entered its final quarter.
The game seemed set for a frustrating conclusion but, with just two minutes to go, substitute scrum-half Paul Marshall was set over the line by his backline colleagues.
Ulster stay fourth with the win and the province will now wait until the end of the month for their next game when they face a trip to face Glasgow on March 25.
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