Both sides scored one try only, but the boot of Wasps' New Zealand-born fullback, Mark Van Gisbergen, saw them through in the end, after a deluge of second-half replacements had helped the visitors, too.
Wasps coach Warren Gatland started the game with prop Craig Dowd, lock Simon Shaw, No.8 Lawrence Dallaglio, scrum-half Rob Howley and wing Josh Lewsey all on the bench, but they came on with about 20 minutes left and all made a difference in the build-up to hooker Trevor Leota's score in the 64th minute.
Wasps had earlier run into a 6-0 lead after just nine minutes, thanks to two Van Gisbergen penalty goals, but Leeds gave them some nervous moments when flanker Colm Rigney powered over from close-in.
Leeds' Scotland fly-half, Duncan Hodge, converted for a 7-6 lead, only for the visitors to reclaim the lead through Leota's score.
Van Gisbergen missed the conversion, but Wasps now have 72 points from 20 matches, with Bath one point ahead after the same amount of games.
Both sides have two more matches remaining this season. Wasps face Gloucester and Saracens away from home, while Bath play Harlequins away and Gloucester at home.
For Leeds Tykes:Try: RigneyCon: Hodge
For London Wasps:Try: LeotaPens: Van Gisbergen 2