Harry Kane put the visitors in front inside four minutes after rising highest to score his 11th goal of the season from Eriksen’s floated corner.
But Swansea responded in excellent fashion to going a goal down and went close to an equaliser on two occasions through the impressive Wilfried Bony.
However, the Ivory Coast international was not to be denied for long as he converted Wayne Routledge’s cross minutes after the restart to score his eighth goal in his last 10 league appareances.
In an entertaining second-half both sides pushed forward for a winner with Gylfi Sigurdsson and Kane coming close to winning all three points for their respective sides.
But it was left to Eriksen to seal victory for Mauricio Pochettino’s side and move them up to seventh in the table, four points adrift of the Champions League places.
Spurs were fast out of the blocks and Kane gave the visitors the perfect start after he put in a determined leap to head home Eriksen’s corner. It was the England U21 striker’s third league goal of the season but Monk would not have been happy with his side’s defensive effort from the set piece.
After the goal Swansea started to dominate possession and their first chance fell to the in-form Bony, but after controlling Routledge’s through-ball on the edge of the box, the Ivorian’s shot was well saved by Hugo Lloris.
Tottenham were seeing less and less of the ball but still represented a danger to the Swansea back four on the break and Kane nearly doubled his side’s advantage with a snap shot from the edge of the box.
However, Swansea were starting come on strong and their best chance in the first half fell to Bony.
If there was anyone the home side would have wanted bearing down on goal in the 23rd minute it would have been the in-form striker, but just as the Ivorian was about to pull the trigger on Jefferson Montero’s through-ball former Swansea star Ben Davies recovered brilliantly to make a last-ditch tackle.
The home side were piling on the pressure now in what was becoming a feisty affair in South Wales – both Ryan Mason and Jan Vertonghen were booked for visitors – and once again it was Kane and Bony at the heart of the action.
First, Kane tried to drag his side forward with a mazy run into the Swansea area. The 21-year-old went to ground with his side claiming a penalty, but referee Robert Madley was on hand to wave away the Spurs appeal.
Then, after brilliant passing between Sigurdsson and Angel Rangel, the Spanish full-back was released into space down the right. His 43rd minute cross found Bony in space but his goal-bound shot was blocked by Fazio.
Spurs were rocking and in desperate need of the half-time break as Routledge made a clever run in behind the Spurs defence but failed to make contact with Neil Taylor’s long pass.
The pattern of play after the interval was in much the same vein as the first half and Swansea were finally rewarded for their efforts. It was that man Bony who did the damage.
Routledge found plenty of space down the right and had all the time in the world to pick out his man. The Ivorian duly delivered, finally beating Lloris after his first attempt was blocked by Fazio.
It was the least the home side deserved and Mauricio Pochettino reacted by bringing on Mousa Dembele for the ineffective Roberto Soldado as his side tried to stem the flow of Swansea’s attack.
Ki Sung-yueng was the next to try his luck at Lloris’ goal but the South Korea international could not direct his glancing header on target.
Despite all Swansea’s possession, Tottenham did enjoy a decent spell of pressure as first Mason tried his luck from long range and then Kane sent a curling shot inches wide of Gerhard Tremmel’s far post.
It was end-to-end stuff at the Liberty Stadium as both side’s searched for a winner and Sigurdsson, who was playing against his former club, curled a left-footed shot into the side netting.
Heading into the final 20 minutes Swansea regained their foothold on proceedings with substitute Jonjo Shelvey and Montero both going close.
But Tottenham finished strong and Kane thought he’d won it for the Pochettino’s men with just over 10 minutes remaining after connecting with another Eriksen corner. This time his header drifted just wide of the far post.
Time was running out for both sides to grab a winner but with just a minute remaining of normal time Eriksen picked up Jazz Richards’ poor clearance and fired a powerful strike past Tremmel from the edge of the box to hand the visitors a dramatic victory.
The win lifts Spurs up into seventh in the race for the top four and extends their unbeaten record over Swansea in the Premier League.
Swansea: Tremmel (6), Rangel (6), Bartley (6), Williams (5), Taylor (6), Britton (5), Ki (6), Routledge (7), Sigurdsson (7), Montero (6), Bony (7).
Subs used: Shelvey (5), Gomis (5), Richards (4).
Tottenham: Lloris (7), Walker (6), Fazio (5), Vertonghen (6), Davies (7), Bentaleb (6), Mason (6), Eriksen (8), Kane (9), Lamela (6), Soldado (5).
Subs used: Chadli (5), Dembele (6), Stambouli (5).
Man of the Match: Harry Kane
Highlights of Swansea City v Tottenham are available to Sky Go subscribers.