"Other teams would have got beaten by 30 points out there, but we kept going. Leinster are a good side and they deserved to win," said Cockerill.
"We had chances, but we didn't take them. In the end, you get what you deserve and we lost.
"We thought we could come to Dublin and win and we had a lot of ball in their territory in the first half. But we lost Louis Deacon in the first half, although full credit to my players in ensuring it was still a real contest with a few minutes to go."
Leinster's Heineken Man of the Match, hooker Richardt Strauss, was full of praise for his fellow front rowers, Irish international props Cian Healy and Mike Ross, and is looking forward to moving forward in the competition.
"We knew we had to come out and work hard because they are a very physical pack and they wanted to try to bully us. We also knew they were a side that would fight to the end and the crowd really kept us going in those final few minutes," said Strauss.
"The coaches did a lot of homework on Leicester before this game and it really paid off."