Tries from Tom Varndell and debutant Ayoola Erinle, coupled with the kicking of Paul Burke and Johne Murphy, gave Tigers the points in the first league game in defence of their Guinness Premiership crown.
"We were concerned that we were playing Bristol away in our first game because they gave us some tough games last year and we have not won here for seven years," acting head coach Cockerill said after applauding his players from the pitch at the final whistle.
"We have a massive amount of respect for Bristol and their coach Richard Hill. They are a good team and I told the players they had to go out and earn it, and I think we earned the win. You have to have belief in your team and I believe in my team."
Tigers came back from 13-10 down at half-time to keep Bristol at bay in the second half and get the season off to a winning start at the Memorial Ground.
"We were a bit frustrated in the first half but I thought we defended very well," Cockerill added. "We gave away some penalties but we did not panic, we stuck to our gameplan and that is credit to the players.
"Maybe the score flattered us a bit in the end but we controlled the game exceptionally well, played it in the right areas of the pitch and we took our chances. We deserved the win. Tactically, the guys were spot-on, the key guys across the backline were excellent and the forwards did a lot of good work.
"I am delighted with the win, but now we have to push on and focus on Bath this weekend.
"Bath got a good win on Saturday. We are looking forward to going back to Welford Road and playing in front of the home support and hoping that our spirit and the tactics are as good again."
Tigers flanker Owen Finegan had extra reason to remember his Tigers debut as his wife gave birth to a baby boy on Saturday afternoon before he caught up with the squad in Bristol.