Leicester led 17-0 after 20 minutes, through a penalty try and a Geordan Murphy score, but they allowed the visitors back into the match and a shock victory for the Sharks was a real possibility as they had much the better of the second half.
Fraser McKenzie and Rob Miller scored tries for them but Leicester, for whom Alesana Tuilagi crossed in between times, eased home with a Toby Flood penalty before Nick Macleod's drop-goal earned Sale a deserved losing bonus point.
Cockerill said afterwards: "We were under the cosh. It's a game we could have easily lost but you can see how much it meant to them by the jubilant reaction on the final whistle at securing a bonus point.
"We would have liked to have got a bonus point but it's a job done, nine points in two league matches over the holiday period with a quick turnaround and seven changes to our side."
Cockerill's one complaint was with referee Andrew Small as he fumed: "The breakdown area today was a farce. We are the team trying to play rugby but they are lying all over the tackle area and it has to be penalised.
"The officials didn't get it right today and a large holiday crowd deserve better."
Sale's performance was admirable considering they lost two players through illness before kick-off and two more during the match.
David Seymour was helped off after a clash of heads within two minutes of the start with team-mate Henry Thomas, who later suffered a leg injury.
Coach Tony Hanks said: "The boys managed so well to deal with two late drop-outs and two early injuries. It will do a lot for their confidence.
"We kept at it and we never gave up, we just ran out of time but we definitely deserved something out of the game."
On the injury victims, he added: "Dave Seymour looked very bad when he came off but he seems okay now. I'm disappointed for the lad as he is the form number seven at the moment and could have displayed his talents on TV.
"Henry Thomas is on crutches, he turned a knee but hopefully it is not as bad as first feared."
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