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Ospreys were down to 14 men on 34 minutes when Ryan Bevington was shown a yellow card.
Dan Biggar missed a drop goal just before half-time, which saw the sides level at 10-10.
There was little between the two sides after the break. Three penalties for each team kept the match level with less than 20 minutes to go.
A busy 5 minutes for Toby Flood saw the fly-half score a penalty, miss a penalty, run in a try and kick the conversion to open up Leicester's first notable lead of the match.
Further tries from Ben Youngs and Manu Tuilagi took the game away from Ospreys.
Despite losing, the Welsh side stay ahead of Leicester in pool 2 on points difference.