Alli’s penalty, after Johannes Eggestein had levelled with six minutes left, looked to have won it – only for Mamoudou Karamoko to curl in late on.
Gareth Bale had earlier also scored from the penalty spot following Peter Michorl’s opener for the Austrians.
Son Heung-min then put Spurs ahead before the late flurry of goals.
Tottenham are through with a game to spare but will need to beat Royal Antwerp at home next week if they are to progress as group winners.
Spurs’ fringe players fail to shine
With the north London derby against Arsenal on Sunday, Jose Mourinho made eight changes to the side that drew 0-0 at Chelsea, but Alli and Harry Winks were still only on the bench.
Those given their chance did little to impress and a sluggish start on a freezing evening in Linz meant a string of saves from Hart were needed to keep the side sitting second in the Austrian Bundesliga at bay.
LASK’s pressure eventually told as the hosts broke following Matt Doherty’s slip and Michorl’s long-distance effort beat former England number one Hart down to his left.
Tottenham struggled to string a pass together before their first move of note saw Tanguy Ndombele’s shot hit Andres Andrade’s arm and Bale converted from the spot for his second Spurs goal since returning on loan from Real Madrid – and the 200th of his career overall.
Two of the three players who kept their places from the goalless draw at Stamford Bridge then combined to put Spurs ahead as Ndombele seized on an error in the LASK midfield to slide Son in on goal.
Bale and Son turned the game in Mourinho’s side’s favour and almost combined for a third, only for the Wales forward to place his pass behind the South Korean.
Alli has found himself out of favour under Mourinho this season, only featuring twice in the Premier League with his game time largely restricted to European nights.
The 24-year-old did get his chance with eight minutes remaining but LASK were level moments later when Eggestein, who hit the post in the first half, beat Hart from the edge of the box.
Alli proved he was able to keep his nerve as he put the visitors in front again from the spot with four minutes to play.
But LASK deserved to take at least a point from the game and they got it when Karamoko fired in a dramatic third in the 93rd minute.
Bale hits 200 goals – the stats
- Spurs conceded three goals in a game for the first time since October’s 3-3 draw with West Ham – they had only conceded three times in their previous nine games before tonight.
- This was only the second time an Austrian side has scored 3+ goals in a major European match against English opponents, with FC Red Bull Salzburg losing 4-3 against Liverpool in last season’s Champions League.
- Son Heung-Min has scored 12 goals in all competitions this season – among Premier League players, only Harry Kane and Dominic Calvert-Lewin (13 each) have more.
- Only Leicester City (8) have scored more penalties in all competitions among Premier League teams than Spurs this season (7).
- Gareth Bale scored his first European goal for Spurs since March 2013 against Inter Milan.
- This was his first away European goal for the club (excluding qualifiers) since his hat-trick at the San Siro against Inter in the Champions League in October 2010.
- Of Bale’s 200 goals, 57 have been scored for Spurs, with the other 143 coming at Southampton (5), Real Madrid (105) and Wales (33).