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Montpellier had the majority of the possession in the first half, with Leicester hampered by the sin-binning of Miles Benjamin, but the hosts turned down several chances to kick at the posts to push for a try, yet failed to find the whitewash.
After returning to the pitch, Benjamin thought that he had broken the deadlock at the start of the second half following a quick move from Ben Youngs, but the replays showed that his foot was in touch, ruling the score out.
Tom Youngs finally managed to add the first points of the game 10 minutes into the second half, barging his way through the French club's defences, but Enzo Selponi cancelled out his effort with a try of his own two minutes later.
Lucas Dupont extended Montpellier's lead further and thought that he had done enough to give the home side a win, but Goneva smashed his way through inside the final minute to give the visitors the victory.
The result leaves Leicester just three points behind Ulster in pool five.