Spurs trailed at the break after Daniel James scored his first Leeds goal since his £25 million move from Manchester United when he finished at the back post, following excellent work from winger Jack Harrison shortly before half time.
Leeds made the brighter start at the Tottenham Hotspur Stadium and went close to taking the lead when Stuart Dallas' fierce drive whistled past the post, a few minutes after Kalvin Phillips had forced Hugo Lloris into a save from a free kick.
The visitors, who were dealing with the absences of several first team players including Raphinha, Rodrigo and Patrick Bamford, were impressive and Spurs struggled to get out of their own half at times.
There were boos from the home supporters at the half-time whistle but Tottenham were much improved in the second period, with Harry Kane and Son Heung-Min both having attempts deflected on to the post within six minutes of the restart.
Hojbjerg pulled Spurs level in the 58th minute as he steered in Lucas Moura's cut-back, as Tottenham's more positive approach was rewarded with their first Premier League goal in four matches.
The home side then survived a late VAR review after Joe Gelhardt's shot struck Kane's arm inside the box, but the Tottenham striker was too close to the shot for it to have been intentional.
The result moves Spurs to within four points of fourth place, after West Ham, Arsenal and Manchester United all lost this weekend, while Leeds remain just above the relegation zone in 17th place with two wins from their opening 12 matches.