The hosts were 15-5 up by half-time despite George Ford's superb solo try and although the England fly-half set up Stuart Hooper for a second score after the break, the Leinstermen ultimately held on.
The win sets up a semi-final trip to Toulon in a fortnight's time and captain Heaslip knows his troops must produce an improved display in that contest.
"In two weeks time, we have got to go to another level," said the skipper. "That's what we talked about pretty quickly after the game.
"A lot of character was shown to stay in the hunt in that last 20 minutes when they had the ball. They are a very hard team to defend against and they just kept coming and coming.
"To hold out and not give a penalty away in that last set showed character. We'll take those positives and work on the negatives as well.
"We're just going to rest up now. A lot of us were saying that's probably one of the fastest, toughest games we've played, including internationals. That's right up there."
As for the Challenge Cup, Edinburgh reproduced their good GUINNESS PRO12 form to hold off London Irish 23-18 at the Madejski Stadium.
Edinburgh raced into a 16-0 lead thanks to the boot of Sam Hidalgo-Clyne and a WP Nel try before Irish fought back with Conor Gilsenan and Scott Steele crossing the whitewash.
But the visitors responded through Fraser McKenzie before the TMO ruled that Exiles hooker David Paice was held up over the line as the clock went red.
And the Scottish side will play Newport Gwent Dragons at BT Murrayfield in the last four after Lyn Jones' men came out on top 25-21 in the all-Welsh battle with Cardiff Blues.
Just one place separates the sides in the GUINNESS PRO12 table and there was little between them at Rodney Parade.
The Blues established a 14-3 lead following scores from Lloyd Williams and Gareth Anscombe before the hosts responded through Hallam Amos.
Nic Cudd then put the Dragons in front before a penalty try sealed the deal, despite Josh Navidi's late effort.
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