We took the lead through Jack Roles at the Ulrich Haberland Stadion - next door to the BayArena - but were ground down by the home side in the second half.
Jakub Bednarczyk levelled on 57 minutes and having seen Tashan Oakley-Boothe rattle the crossbar, Leverkusen forged ahead via Anton Walkes' own goal 10 minutes later.
It stayed that way until added time when Tim Handwerker was bundled over by left-back Jaden Brown and picked himself up to convert from the penalty spot.
The result means we're yet to put any points on the board after three games in Group E.