The Tigers have found life tough in the opening weeks of the season owing to a number of World Cup call-ups, but they dug deep when it mattered most to claim only their second Premiership win of the season.
Rookie wing Tom Brady's first-half try helped the Sharks into a 13-3 interval lead but second-half scores from Alesana Tuilagi, Martin Castrogiovanni, Toby Flood and Marcos Ayerza completed a stunning turnaround.
Asked about his side's fightback, which secured a bonus point, Cockerill said: "Historically we've been very good at that.
"We believe in the systems that we use conditioning-wise.
"We've had a bit of stick across the board about everything in the past six or seven weeks, but we've stuck to our guns and we've worked through it and believed in the stuff we do.
"Tonight we got proved right, but it's one win and there's a long way to go.
"We've got London Irish next week, who are a very good side, and we need to be more accurate in certain parts of our game because in other games you will get punished and not recover from that."
Castrogiovanni's introduction at the expense of England prop Dan Cole during the second half proved crucial as he scored one try and kept Leicester on the front foot.
Cockerill added: "Coley and Castro are two world-class tightheads and they both wanted to play.
"The two most important positions are your tighthead and your reserve tighthead for me and they proved that tonight."
Sale executive director of sport Steve Diamond was left deeply frustrated by his side's second-half capitulation.
The reverse served as a reality check for the Sharks and ended their winning start to the season on home soil after three previous victories.
Diamond said: "I think after 10 minutes of the second half we realised where we are at.
"We had no ball and the pressure told. Against a side like Leicester you can't compete.
"Our set-piece let us down, they scored from two driving mauls five metres out and our scrum didn't look very stable for most of the game.
"We've got a lot to work on and we got taught a lesson today, but that's how you learn.
"We will go away and lick our wounds and come back. We showed how good we can be in the first half but the second half we showed our Jekyll and Hyde character."
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