Cockerill and the team were given a huge reception by a 20,000-plus crowd at Welford Road as tries from Alesana Tuilagi and Steve Mafi plus 14 points from young fly-half George Ford, earned a place in the final for an eighth successive year.
“I’m just delighted for all the guys,” said director of rugby Cockerill.
“We’ve had a lot of stuff thrown at us over the season and it’s been hard under the circumstances at times. But now we’re in the Final for the eighth year on the trot so we’re doing something well.
“I’ve learned that you should savour days like this because they are special. But we want to go there and try to win it now.”
Tigers and Saracens had battled out the previous two Premiership Finals, with one win each, and there was little to separate them at Welford Road until Mafi’s second-half try created a five-point lead and George Ford quickly added a 52nd-minute penalty to put the hosts more than one score ahead.
“It was tight, just as we knew it would be,” said Cockerill. “We chose to play into the stiff breeze in the first half and I was happy to be a point up at half-time.
“We would have liked a bit better control of field position in the second half and we missed a couple of penalties, but we scored two tries to nil and I thought we were good enough got the win in the end.
“Three years ago when Saracens made all their changes everybody raised an eyebrow but they are a very good side, no doubt about that. They are the defending champions of England. They’ve got some serious guys missing through injury too. They are a very good side, but so are we and today it was our turn to win.”
Harlequins came through the other semi-final with a late score to beat Northampton Saints 25-23 and now meet Tigers at Twickenham on May 26.
“Quins finished the league season on top and they have shown some resilience – to be five points down with just a few minutes left in a semi-final against a good team shows that,” said Cockerill.
“The way both sides play you would expect a good game.”
Tickets for the Aviva Premiership Final are now on sale to Tigers season ticket holders and members, with general sales open from Wednesday.