Ian Madigan and Andy Goode both kicked over penalties early on for their respective sides, although it was not until the 24-minute mark that the game had its first try, with Alapati Leiua crossing over to put the visitors ahead.
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That lead did not last long, though, as Darragh Fanning opened his side's try count just short of the half-hour mark when he found the chalk.
The momentum of the game swung once more when Christian Wade touched down five minutes before the interval, yet Wasps could not hold onto their advantage following the restart.
Leinster came out the stronger side, adding two tries in the space of eight minutes through Fanning and Dominic Ryan to edge ahead in the Irish capital.
There was still 20 minutes remaining for Wasps to find a response, but it was not to be as Les Bleus held out for the points in Pool Three.