Sixteen unanswered second-half points gave the Premiership champions the perfect start to the defence of their crown and ended Bristol's record of never being beaten in a Premiership opener.
Cockerill was thrilled to have claimed victory saying: "We have not won here in seven years and it is a fantastic start for us and I can't hide my satisfaction with that. Bristol are a very good team and I have a huge respect for Richard Hill as a coach for obvious reasons.
"I knew that if we played to our potential and we played tactically the right way then we could certainly get a win. The scoreline maybe flattered us a little but the boys controlled the ball exceptionally well. We played in the right areas of the field and we took our chances.
"Fair play to our lads in the end everybody thought that we deserved to win.
"The way that the referee was controlling the game if you had the ball and kept hold of it you were going to get some penalties and be rewarded for that. We'd done our homework on the referee and Bristol.
"We scored a relatively opportunist try in the first half through Tom Varndell which he finished very well. There was some great work from our forwards which nullified Bristol pretty much for large periods of the game. We have worked more on our defence in pre-season with [assistant coaches] Andy Key and Neil Back doing good jobs.
"I have a huge amount of respect for the Bristol team. They are a good side and this is a hard place to come and play. We have lost here with some very big names in our side and today, for some of our guys it was a great start of their debuts.
"Also a debut win is a great start to the season when we are missing so many players - it is fantastic. We had tried to be quite smart with our recruitment but we have virtually a whole pack of forwards to come back [from the World Cup] and I am looking forward to having those guys back."