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Leicester also left it until the last 30 seconds to steal a 15-14 victory at Montpellier.
After a scoreless first half, the Tigers struck first through Tom Youngs, but went behind to Enzo Selponi’s converted score just two minutes later.
And it looked like defeat was sealed with Lucas Dupont’s 62nd minute score that put the hosts up 14-5.
But Ryan Lamb’s penalty kept the Tigers in touch, and when Niki Goneva was put away by Marcos Ayerza in the 79th minute, he had the presence of mind to get round under the posts for Lamb’s winning conversion to be a simple one.
Leicester now have their fate in their own hands, with Ulster travelling to Welford Road in the final round in what could be a tasty pool decider.