It looked as though Dele Alli's penalty would seal an undeserved 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.
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 first half and Son Heung-min put Spurs ahead soon after the restart.
Johannes Eggestein levelled with six minutes remaining but there was still time for Alli - on the field for just four minutes - 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.
Hart had already denied Eggestein and Rene Renner before Michorl arrowed a shot beyond the former England goalkeeper to give Dominik Thalhammer's men the lead, though that soon disappeared.
Andres Andrade handled in the box in first-half stoppage time and Bale coolly converted from the spot to level matters against the run of play, bringing up his 200th career goal.
Tottenham were scarcely any more polished at the start of the second half, but one moment of quality told as Tanguy Ndombele's excellent throughball picked out Son to provide an assured finish past Alexander Schlager.
It was hard on LASK but they fought back to level when Eggestein's shot somehow eluded Hart, whose blushes looked to be spared when Alli promptly netted his spot-kick at the other end.
But Hart was beaten again by Karamoko to give LASK a point that was the least their efforts merited.
What does it mean? Top honours up for grabs
Top spot in the group is still to be settled, with Royal Antwerp in the driving seat after they beat Ludogorets 3-1.
This is not exactly the preparation Mourinho might have wanted for the forthcoming showdown with Arsenal, but at least he was able to rest a number of players.
For LASK, it represents a valiant way for their Europa League hopes to come to an end.
Fringe players fail to stake a claim
Given the magnitude of the game that lies ahead for Spurs in a few days, you might have thought that players on the periphery would have been keen to make an impact and force their way into Mourinho's plans.
But this was a poor showing practically across the board, with Spurs managing just three shots on target and finishing with a passing accuracy of 66 per cent.
LASK also had 54 per cent of possession, further highlighting how they were simply the better side.
Eggestein a constant threat
Even before his goal, Eggestein caught the eye. He had three shots, created one chance and had an 82 per cent pass completion percentage from his 38 passes.
10y 44d - Excluding qualifiers, Gareth Bale's penalty for Spurs was his first away goal in UEFA European competition for the club since scoring a hat-trick at the San Siro against Inter Milan in the Champions League in October 2010, 10 years & 44 days ago. Equaliser. pic.twitter.com/drmtwyHzmL— OptaJoe (@OptaJoe) December 3, 2020
Key Opta facts
- Spurs conceded three goals in a game for the first time since October's 3-3 draw with West Ham in the Premier League - they had only conceded three times in their previous nine games before today.
- This was only the second time an Austrian side has scored three-or-more goals in a major UEFA European match against English opponents, with Salzburg losing 4-3 v Liverpool in last season's Champions League.
- Son has scored 12 goals in all competitions for Spurs this season - among Premier League players, only Harry Kane and Dominic Calvert-Lewin (13 each) have more in 2020-21.
- Only Leicester City (eight) have scored more penalties in all competitions among Premier League teams than Spurs this season (seven).
- Bale scored his first European goal for Spurs since March 2013 v Inter, while this was his first away European goal for the club (excl. qualifiers) since his hat-trick at the San Siro against Inter in the Champions League in October 2010.
Arsenal visit Tottenham Hotspur Stadium on Sunday before Spurs conclude their Europa League group campaign at home to Royal Antwerp, on the same day LASK head to Ludogorets.