The reigning GUINNESS PRO12 champions found themselves 20-11 down to Wasps at half time at the RDS but second-half tries from Darragh Fanning - his second of the game - and Dominic Ryan helped them to a 25-20 win.
But despite shutting out the visitors after the break, captain Heaslip expects Leinster to learn from their first-half errors this week.
"They punished us for any mistakes we made and took their scores really well," he said.
"We always knew they are a team that has unbelievable gas, so we had to close down the space, as well as their big ball-carriers in (Nathan) Hughes and Ashley (Johnson).
"We had our work cut out for us but I think we worked our scores really well and it was a huge effort by everyone, especially when we got into the 22.
"We knew a lot of our own mistakes killed us in that first half and Wasps were punishing us for those mistakes and they were controlling the ball very well.
"I thought our defence was good but we have a lot to work on, especially with Castres coming up next week."
Earlier in the day Scarlets made two-time reigning European champions Toulon work hard for victory in France.
John Barclay and Kristian Phillips each crossed over for Scarlets but the French outfit were ultimately too strong as former Cardiff Blues full-back Leigh Halfpenny kicked 13 points to take Toulon to a 28-18 win.
Ospreys ran in five tries to get their European campaign off to a flyer against fellow GUINNESS PRO12 outfit Benetton Treviso.
The Italians are still without a league win this season and things didn't get much better at the Liberty stadium as Jeff Hassler, Rhys Webb and Dan Baker each crossed over in the 42-7 victory.
Full-back Dan Evans bagged himself a pair of tries and Dan Biggar kicked 17 points from the tee, while Ludovico Nitoglia's late converted score was simply a consolation for the visitors.
GUINNESS PRO12 high-flyers Glasgow Warriors produced a magnificent demolition of Bath, welcoming the in-form Premiership outfit to Scotstoun and romping to a 37-10 triumph.
Mark Bennett crossed over first and last for the hosts, while Sean Maitland, Tommy Seymour and Niko Matawalu also dotted down - Bath managed just one try, through Jonathan Joseph.
And Ian Keatley left it late to clinch victory for Munster, sending a last-gasp drop-goal over to see off Sale Sharks 27-26.
Munster were 23-7 down at the break but Andrew Conway and Conor Murray added to Dave Kilcoyne's early try before fly-half Keatley finished the job.
"He showed great composure," said head coach Anthony Foley. "I thought he struck it sweetly.
"He was very good defensively, too. He's growing into the ten shirt, which is a big shirt to wear, and it's great to see him pull out a performance like that away from home."
Ulster were beaten 25-18 at Leicester Tigers, as second-half tries from Tommy Bowe and Franco van der Merwe were not enough to overturn a 16-point deficit at the break.
Edinburgh ground out a 15-13 win away at in-form French outfit Bordeaux-Begles on Friday night, but there was no such joy for Zebre, victims of a second-half fightback in 33-24 defeat to Oyonnax.
Sam Warburton got on the scoresheet as Cardifff Blues beat Grenoble 37-14 at BT Sport Cardiff Arms Park, while fly-half Rhys Patchell racked up 20 points of his own.
Robbie Henshaw scored two of Connacht's eight tries as they thrashed La Rochelle 48-12 - a far more clinical victory than their 9-6 triumph over Treviso last time out in the GUINNESS PRO12.
And there was a memorable win for Newport Gwent Dragons, who returned from the French capital with a 38-22 bonus-point win over Stade Francais, Richie Rees, Rhys Thomas, Hallam Amos and Nic Cudd each crossing the whitewash.
Dragons captain Lee Bryne said: "Some of us, including myself, might not have a chance to come here again and get a win like that.
"I never did while I was with Clermont and I'll probably never have a chance to come back here. I'm so happy for the young players."
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