Alex King, off target four times early on, found his range and kicked 15 points to put Wasps out of sight, while Andy Goode kicked three penalties for the Tigers.
Dallaglio said: "Everyone was talking about the Leicester guys but when you're underdogs it pumps you up.
"We knew if we put Leicester under pressure, then it would be a tough game and everything worked really well.
"Everyone, to a man, was outstanding - Leicester are a great team and we feel very proud to have beaten them on the greatest stage of all."
The 34-year-old number eight also hit back at critics of his side's form this season.
"People say the lineout hasn't worked well all season, well it worked pretty well today," he added.
"People say the scrum hasn't worked well all season, I'd say it worked well today."
For Leicester coach Pat Howard, in charge of his last game with the club, it was an unhappy send off.
But despite failing to complete an unprecedented treble, he insisted he could be proud of what has been an outstanding season.
"Wasps deserved their victory. We weren't allowed to play and we lost as a result.," he said.
"It was always possible, they are a very physical side and that's where they won it today.
"There were two well-taken tries but we had opportunities and didn't take them.
"I know we were good enough to win it but we didn't play well enough. To do the treble is an amazingly hard thing. You have to turn up a lot of times and this is the first time we haven't.
"It's been pretty special this year, but we've let ourselves down today. I'll be down for an hour or so then I'm going to celebrate our two wins."
Dallaglio's opposite number, former England captain Martin Corry, echoed Howard's sentiments.
"The first half hour was the key. Hats off to them, they were better than us. We set ourselves high standards," he said.
"But when I look back I'm hugely proud - of wearing the shirt and of everyone representing the side, from the top man down to the bottom.
"It's a season to be proud of."
Wasps rugby director Ian McGeechan insisted he had wanted to see if Leicester would be "frightened" by the intensity of his team early on.
"The idea was to get stuck into the final," said McGeechan. "We talked about the first 10 to 15 minutes and said we had to go full out and play at the level they had played at for the last few weeks. We wanted them to be frightened by the intensity of it. That got us into the game.
"We mixed the game up, not doing some of the things we've been doing and it worked well. It was a case of trying to mix and match when we were playing with the ball and when we were without it. We wanted to be as cute as we could be."
He added: "There is a great mix of old and young and that told - the experience and the vibrancy in the team."