An Alesana Tuilagi try in the last 10 minutes of a tight and tense semi-final encounter at Welford Road eventually secured an 11-3 victory and a Twickenham final against Saracens on May 28.
“We have not been in particularly brilliant form in the last few weeks but I was very pleased with our control of field position and control of the ball against a really good side,” said director of rugby Cockerill.
“A lot of people had written us off today but don’t under estimate this group of players.
“Saints have been in good form and they are in a European final, but our players deserve a huge amount of credit for fronting up and getting the win.
“It was a really good contest and we never really let them get on to the front foot, which was very important.”
Tigers led 6-3 as the game entered its last 10 minutes when Tuilagi scored his 17th try of the season after smart work on the left wing by front-rowers George Chuter and Marcos Ayerza.
Cockerill said: “We were just three points up and anything can happen so the try came at a crucial time.
“We led 3-0 at half-time and thought there was probably more of an advantage than that, but it helped to focus minds again for the second half. We controlled the ball for large amounts of time and our attitude to defend and not let them get any cheap points out of the game was excellent.
“The team have reacted to a bit of stick with a very, very good performance, and Toby and Ben controlled the field really well. It was our best all-round team performance for a while.”
Tigers now meet Saracens, conquerors of Gloucester in the other semi-final, in a rematch of last year’s Premiership Final.