Christian Eriksen scored a stoppage-time winner to break Burnley's brave resistance as Tottenham kept track with Liverpool and Manchester City.
Eriksen's strike in the 91st minute broke Burnley hearts with a powerful shot that rifled past Joe Hart.
The victory has moved third-placed Spurs to within three points off Liverpool and Burnley remain down in 17th just two points above the drop-zone.
How It Happened
It was a game of very limited clear-cut chances for the home side - who were perhaps a tad fortunate to take the three points in the end.
Lucas Moura had the game's first chance after he was found by Erik Lamela; but the Brazilian winger couldn't guide his shot on target.
Oliver Skipp had a headed chance just before half-time when he was found by Kieran Trippier with a good cross.
Ashley Barnes had the best Burnley chance of the opening 45 minutes, but his header couldn't be kept down from the centre of the box.
Lamela forced a save out of Hart with just over 15 minutes remaining after being found by Danny Rose.
The Argentinian had another chance for the hosts - but he couldn't find the target on this occasion.
It looked as if Spurs were going to drop two points at Wembley when Dele Alli saw yet another chance go begging for the home side.
However, Eriksen had a different idea. Harry Kane laid the ball off to the Denmark international and he slammed past Hart to win the game for Spurs in dramatic style.
The Managers
Tottenham Hotspur boss Mauricio Pochettino described the late win today as "fantastic" reflecting on a good week for his team.
"It is a very good week for us and a fantastic win for us today," Pochettino said. "It is so important to be focused and try to keep our energy.
"After the Barcelona game and qualifying for the next stage of the Champions League, it is difficult to find the way to perform like this.
"Today I congratulate my players' effort and the fans. It was a late victory but so important for us."
Burnley manager Sean Dyche has said he "feels for his players" after seeing all their hard work thrown away in the last minute of the match.
"Obviously we are frustrated and disappointed in the outcome but a lot of positives," Dyche admitted. "My only gripe is we get done by a whack down the pitch.
"Inevitably we didn't deal with that final moment and they nick a goal. There were good signs from the mentality we know we have to win games.
"There is a resilience and organisation but we are playing a top side - it is difficult to come down here.
"Away from home against sides like this you have got to try and model a team that can still get a result.
"I feel for the players because they put a proper shift in today and did the basics well. The last three or four games we are getting back to what we can be - we need more attacking prowess but is difficult."