The twice European champions were handed a home tie against the Ospreys in the opening round, giving them a chance to avenge their EDF Energy Cup final defeat to the Welsh region last season.
Leicester, who have a new coach in South African Heyneke Meyer, then face a trip to Italian minnows before another home clash with French giants Perpignan, who they also meet away six days later.
In 2009, Meyer's side then host Treviso before an away trip to Swansea to meet the Ospreys again.
"The way that the fixtures have fallen for us is quite nice," Cockerill told the Leicester Mercury.
"It's a good opportunity to get off to a strong start and, with Treviso in between the home games, there's a good chance to win the first three.
"The back-to-back games are against Perpignan and the first one is at home.
"I would like to think that we will go there for the fourth game with the possibility of them being in a position where they may be struggling to qualify.
"The draw has been pretty favourable to us."
Leicester failed to progress from a difficult group in 2007-8 in what was a disappointing season that culminated in their head coach Marcelo Loffreda losing his job.
The Tigers have also lost their defence coach Neil Back to relegated Leeds and long-serviing fly-half Andy Goode to Brive.
But joining the Midlands club is England back Toby Flood and South Africa international fly-half Derick Hougaard..