Julian Salvi's try double underpinned an emphatic first-half for Leicester, while Manu Tuilagi and Matt Smith touched down, with George Ford kicking four conversions, including the extras to a penalty try. But Treviso scored tries of their own by Dean Budd, Lorenzo Cittadini and Christian Loamanu. Alberto Di Bernardo landed two conversions and two penalties.
"We were inaccurate - we went to sleep in the second half," Cockerill said. "It was a lot closer than we would have liked, but it's job done, it is five points."
He added: "It certainly won't lead us to be complacent next week, so maybe it has done us a favour. We will have to be a lot better. Treviso were good, but I don't think we were under any illusions. Two years ago, we nearly lost in Treviso in the first round of the pool stage.
"The players will know they are in for a tough afternoon next week. We have got to go there and try to get five points. Whether that is attainable, we will have to see. It was job done, but no more than that.
"Five points next week would be very helpful, but after (this) performance from Treviso, in particular, we need to understand the task ahead for us."
Cockerill, meanwhile, paid tribute to Australian Salvi's display, adding: "He has been very good for us. With Ben Woods and Craig Newby both retiring, it left us a little bit short in the seven area, and Julian is a very good player."
Salvi admitted the second period had prompted some "clock-watching" in the Leicester ranks, yet he also backed the Tigers to learn from their latest European experience.
"There was a bit of clock-watching at one stage to see how close they (Treviso) were getting, but we are strong enough and experienced enough as a team to make sure we get the wins, especially at Welford Road," Salvi said.
"You need to give credit where credit is due, and Treviso came out well in the second half. They made us work and started to grind away through the middle."
- Sports & Recreation
- Julian Salvi