Despite being 23-6 behind at half-time, the defending champions got within one score of their great rivals midway through the second half, but it was Wasps who maintained their lead to win the title.
We went out expecting and believing we could win, said captain Martin Corry who was chasing his sixth Premiership crown in 11 years at the club.
We had a gameplan to do it but the execution, especially in the first half, was not there. We know how good we can be, the frustrating thing is we did not do that for 40 minutes.
When we show how well we can play we are a very good side, but you cannot give a side as good as Wasps a head-start in a final.
At half-time we still had belief, we still thought we could win it. You believe that up to the final whistle.
A world record crowd for a club match of 81,600 packed into Twickenham, and Corry reserved special praise for the supporters.
We have immense support at Leicester Tigers, he said. We are incredibly proud of going out to play for the jersey and the tradition of the club, but also for the support.
Every time we run out you are always incredibly grateful for the crowd. It is one of the biggest disappointments when you lose games like this that you do not give the crowd something to celebrate.
Despite the huge disappointment of being unable to end the season on a winning note, Corry said there is hope for the future.
We have a lot of young, talented players coming through, he said. Guys like Tom Croft and Jordan Crane are coming through, and the front rowers Ben Kayser, Marcos Ayerza, Castro, they are young guys putting pressure on. Full credit to the coaches and the academy that these guys are coming through.
Head coach Marcelo Loffreda, after his second Twickenham final of the season, said: Wasps put us under a lot of pressure and got on the front foot in the first half. That makes it difficult.
I knew we were going to come back in the second half because I know the players. I knew they were going to put in another big effort to do that.
When you get to finals you want to win them so I am very sad and hurt for the team but now we have to look forward.