The result in Austria ensured that Antwerp will also advance from Group J.
Elsewhere in the early games, Braga, AC Milan, Lille, Villarreal, Dinamo Zagreb, and Red Star Belgrade ensured their places in the last 32 with one round of group games still to play.
They join Leicester, Arsenal, Hoffenheim, and Roma who had already qualified.
In Linz, the home team dominated the first half but only scored once, through Peter Michorl. Spurs, who had not managed a shot on target, won a penalty for handball in added time. Gareth Bale converted for his 200th career goal.
Son Heung-min put Spurs ahead on the counter attack in the 56th minute.
With Linz still threatening, manager Jose Mourinho made a series of defensive substitutions.
Tottenham had three centre backs, three full backs and two defensive midfielders on the field by the 84th minute when Johannes Eggestein still found a huge hole and lashed in a shot that Joe Hart reached but could not keep out.
Three minutes later, Spurs regained the lead. One of the two attacking players, Steven Bergwijn was tripped to earn a penalty and the other, Dele Alli, swept it home.
Spurs could not keep the lead. Young Frenchman Mamoudou Karamoko smashed in an equaliser from outside the box in added time.
The draw was enough to put Tottenham through and next week they will host Antwerp, who beat Ludogorets 3-1, with first place at stake.