Bristol boss Pat Lam was frustrated by injuries and poor discipline for a defeat which leaves his team struggling to make the play-offs.
“It's going to be difficult and it will go down to the wire,” he said. “We've only ourselves to blame for our poor discipline though the scoreline does not reflect the chances we had.
“We showed glimpses of what we could do but we put ourselves under pressure through our own mistakes.”
Bristol were also hit by a late rejig when lock Joe Joyce hurt an elbow in the warm up. That was followed by flanker Sam Lewis being helped off after six minutes with a leg problem.
Leicester boss Richard Wigglesworth said: “I'm just telling the players to concentrate on the next match … if we do that we will make the play-offs.”
Wigglesworth said it was always his intention not to start his England players following their Six Nations efforts. He also revealed full-back Freddie Steward was set to be rested completely but answered a late call to be on the bench when Chris Ashton jarred a leg in training.
