It looked as though Dele Alli's penalty would seal a scarcely deserved three points for Spurs to send them into Sunday's north London derby on the back of a win.
But fellow substitute Karamoko had other ideas and sent a fine strike beyond Joe Hart from 20 yards to make sure the spoils were shared in a strange but memorable Group J clash.
Jose Mourinho made eight changes from the 0-0 draw against Chelsea with an eye on the visit of Arsenal, with Harry Kane a notable absentee as the Premier League leaders toiled to a draw in Linz.
Spurs beat LASK 3-0 in the reverse fixture in October but were more laboured on Thursday, struggling to find their rhythm and falling behind to Peter Michorl's excellent 42nd-minute opener.
However, Gareth Bale levelled from the spot with the last kick of the half and Son Heung-min put Spurs ahead early in the second half.
Johannes Eggestein levelled with six minutes remaining but there was still time for substitute Alli to make his mark from 12 yards after Steven Bergwijn was brought down by Philipp Wiesinger.
Remarkably, that was not the end of it and Karamoko fired home in the dying moments to make it 3-3.