Trailing by 17 points heading into the final quarter, Leinster stayed in it with a try from Sean Cronin but couldn't get any closer in their comeback attempt, and skipper Heaslip singled out the Connacht defence for special praise.
"I don't think it got away from us, the better side won," said Heaslip. "You can't detract from how they got their scores. They were the better team.
"They were going very well this season, they were the pacesetters for the season. They're a tough side to defend against, they punish you in all sorts of ways and they're a very good defensive side, they probably don't get the credit they deserve for their defence.
"There's not a whole lot for me to say, I personally thought they were favourites coming into it because of the way they played all year."
Meanwhile Leinster coach Leo Cullen was disappointed that his team could not reproduce the same display they managed against Ulster in the semi-final.
However he was also quick to praise Connacht for their performance in Edinburgh.
"I'm pretty gutted," said Cullen. "We have a huge amount of respect for Connacht and what they've achieved.
"It's a pretty remarkable season they've had and we have no complaints. We weren't good enough, which is probably the most disappointing thing.
"We didn't perform to the levels we're capable of. A huge amount of credit to Connacht; it was an amazing atmosphere out there and all the support they had is incredible. It's an incredible story for them and they fully deserved the title.
"We just misfired a little bit around the set-piece and some of our ball control, particularly in that first half. We just coughed the ball up unbelievably cheaply. Connacht, on the flip side, were incredibly clinical in taking the opportunities they had."
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