The hosts raced to an 11-3 lead courtesy of a Shane Geraghty try, but a series of penalties and a try from Christian Wade saw Wasps edge ahead just before half time, taking a 16-14 lead into the break.
The Exiles dropped down to 14 men at the start of the second half when Chris Hala'ufia was sent to the sin bin, allowing Stephen Jones to kick another penalty for the visitors.
Marland Yarde put the hosts back ahead once the side was back to 15, before two penalties from Tom Homer extended their lead.
Wasps made a series of changes as they looked to reduce their opponents lead, but they were unable to break the try line again.
The win moves Irish up to 10th, while Wasps remain fifth in the table.