Alberto Paloschi gave the visitors a surprise early lead at White Hart Lane and for 70 minutes it looked like that might be enough as Swansea keeper Lukasz Fabianksi made a string of superb saves.
Spurs substitute Nacer Chadli, however, finally found an equaliser and seven minutes later Danny Rose scored the winner to complete a dramatic turnaround.
The victory means Tottenham reduce the gap behind Leicester back to two points and move three clear of Arsenal, who lost at Manchester United. Swansea sit 16th.
Harry Kane wore a face mask to protect his injured nose, which was broken last weekend against Crystal Palace, but the England striker ceded the limelight here to his team-mate Christian Eriksen, who was a constant presence in Spurs' 34 attempts on goal.
Swansea should have been ahead within the first two minutes as Paloschi picked out Gylfi Sigurdsson with a cutback but the former Spurs midfielder was denied from eight yards by an excellent Hugo Lloris save.
Relieved to be level, Tottenham began asserting their authority and twice cut their opponents open out wide as Erik Lamela lifted Rose's pass over before Kane diverted Eriksen's cross, but was denied at the near post.
Loose touches and misplaced passes had Pochettino doing pirouettes on the touchline and his anxiety was proven justified as Swansea took a surprise lead in the 19th minute.
There was a hint of fortune about the goal as Angel Rangel's shot from the edge of the area deflected into the path of Paloschi but the Italian did well to steer his half volley past Lloris for his first goal in a Swansea shirt.
The home side responded, swarming forward in search of an equaliser, but Fabianski would not be beaten as Eriksen, Son Heung-min and Kane were all thwarted.
In the middle of the Swansea keeper's heroics, Dele Alli thought he had a strong case for a penalty when he tumbled under pressure from Sung-Yueng Ki, but referee Mike Jones shook his head.
There was still time for one more Fabianski stop before the interval and it was arguably the best of the bunch, as Rose's pass found Eric Dier six yards out only for the England midfielder's finish to be blocked by the Pole's right hand.
If Spurs had hoped half-time might break the Fabianski spell they would soon be disappointed as Eriksen's curling free-kick was clawed away and Kane's long shot was pushed around the post.
Rose went even closer with a stinging drive from range but it was the last of Fabianski's resistance as Tottenham finally found an equaliser from the resulting corner, substitute Chadli turning Kyle Walker's shot home from close range.
The roar of relief from the crowd had barely subsided when the hosts scored again as Eriksen's corner was headed out to Rose, who took one touch before firing low under Fabianski's body and into the net.
There were some nervy moments late on for the home fans as Lloris was called upon once more to deny Paloschi's angled volley but Spurs held on for a gutsy win that keeps their title challenge on track.
Source: PA