Despite being 8-6 down at half-time, through Rob Miller's try and Nick Macleod's penalty, Leicester came back in the second period with Ben Youngs' touchdown and the boot of George Ford.
Although Cockerill was delighted with the performance, he also commended the Sharks for their resistance.
"I thought Sale played really, really well and put us under a huge amount of pressure, and we dealt with that sin-binning in the first half," he said.
"We thought it was a pretty soft try to concede, but we were under pressure the whole time and didn't have any ball in the first half.
"They were very good for their lead at half-time, but I thought our boys kept their composure. It's a good win as they were obviously desperate for the win.
"Sale played really well and we dealt with that pressure. Our commitment was good defensively and was as good as we've been for a long time.
"Our discipline to stay onside and roll away, and to deal with all those things was very good. It probably just got us enough ahead to win it."
Cockerill also acclaimed Ford's performance as the fly-half kicked five penalties in his first start of the season.
He added: "I thought Fordy did well. He coped with their physical midfield. I wanted him to play 80 so he could deal with those issues and he did it really well.
"He kicked his goals and kept us in their half in the second period with some sensible kicking."
As for Sale director of rugby Bryan Redpath, it was yet another defeat which continued their disappointing early season form.
However, despite the loss, Redpath thought that there were plenty of positives to build on.
"I'm a lot happier," he said. "There were just a couple of crucial passes again that didn't happen. We needed to get that try just before half-time and build a little bit of a cushion.
"We knew they had a strong bench and we knew we would be under the cosh a little bit in the second half.
"We just had a bad start to the second and never really recovered. George Ford for them just kept kicking them into positions and that made it harder.
"When we raised the tempo of the game, we looked pretty dangerous but, against Leicester, you have to maintain it and we didn't."
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