And Cockerill, who leads Leicester into their fifth Heineken Cup Final since 1997, is confident that the 10-9 win in England's showpiece event could prove the perfect preparation for the final game of the season.
"In some ways, not playing as well as we would have liked has helped galvanise us a little bit more," said Cockerill, who stepped up from his role as assistant coach at Welford Road when previous incumbent Heyneke Meyer returned to South Africa mid-way through the current campaign.
"You win by a hatful at the weekend and suddenly it's a bit too easy, so we've refocused our minds very quickly and the guys are a little bit worried about keeping their spots because they've not played as well as they'd have liked. That's a good reminder for everybody."
Having finished top of the Guinness Premiership standings and then swept aside Bath Rugby in their semi-final, Leicester went into the clash with the Exiles as heavy favourites. They nearly came unstuck, however, as Irish disrupted the Tigers' new expansive style of play and prevented them from getting into their usual rhythm.
"We didn't play as well as we would have liked with ball in hand because Irish got into our ball on the floor," explained Cockerill.
"And our lineout and scrum was a bit sloppy. It wasn't good enough for us but you've got to take the win at the end. We'll take that win, move on and we've just got to be better this weekend.
"Winning that game helps. You've won a tight game when you've been the best team across the board over the season. We've got a trophy and that's great and you have to take satisfaction in that.
"But you can't now say, 'well, we've won a trophy so if we go to Edinburgh and we lose, it doesn't really matter', because I'm not having that. We need to get focused and we need to be right because you don't get too many opportunities to win a double.
"In isolation both tournaments are great achievements but to win them together would be enormous for everybody involved here."
Cockerill knows that achieving such a feat will take a monumental effort against a Leinster side who crushed tournament favourites Munster in the last four just three weeks ago.
Two players at the heart of that Leinster triumph are well known to Cockerill and the whole of Leicester, with second row Leo Cullen and openside flanker Shane Jennings having earned huge reputations during a two-year spell in the East Midlands between 2005 and 2007.
"Leo's captain of their side and he runs their lineout," added Cockerill.
"He captained the side here and for a non-Englishman to captain Leicester is a big thing. That shows you the status he had here.
"I think a lot of Leo as a player - I recruited him and Pat (former head coach Pat Howard) recruited Jenno. Jenno's a real dog on the floor and he'll make the breakdown a mess for us.
"I've got a huge amount of respect for both of them."