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Laurence Pearce went over twice within 10 minutes of the restart to put Leicester back in front as the hosts tried to avoid becoming the first team to lose to the London side this season.
Welsh's hopes of another comeback were dealt a blow when Chris Hala'ufia went in high on Pearce, earning himself a red card.
With the visitors playing the remainder of the match a man short, Leicester were able to dominant the game, adding three further tries in the final 15 minutes to secure a comfortable victory.
The result moves Leicester back into the top four, while Welsh remain bottom with no wins so far this season.