Mauricio Pochettino lost both Victor Wanyama and Harry Winks to injury before the break, stretching the north Londoners' squad to its limits.
But composed finishes from Eric Dier and Heung-min Son ensured Spurs became only the fourth side to win on Burnley's patch this season.
Chelsea suffered a surprise 2-1 defeat to Crystal Palace and the title race is very much still alive, with Spurs proving that they can cope without talisman Harry Kane even in the most testing of circumstances.
Vincent Janssen made his first Premier League start since November and Spurs were determined to settle into their passing game early on, despite Burnley's fierce pressing tactics.
Spurs needed to be alert to keep out a dangerous Matt Lowton cross after Jan Vertonghen was caught in possession but the hosts were guilty of careless errors during the opening stages too and Joey Barton's loose pass gave Christian Eriksen an opportunity to break clear before the Dane saw his effort on goal blocked.
Toby Alderweireld's powerful long-range free kick was the sum of the visitors' early attacking threat while Burnley grew into the game and began to turn the screw with a series of corners.
Despite struggling to summon a creative spark, the best chance of the first half fell to Dele Alli after Eriksen's left-footed drive forced Tom Heaton into a fine save.
The ball span into Alli's path just outside the six-yard box but the 20-year-old fired wide with the goal at his mercy. Even with the close attention of two defenders, the midfielder's reaction suggested he knew he should have done better.
Spurs struggled to control the midfield battle and they lost both of their central midfielders to injury before the break.
Victor Wanyama was withdrawn due to a knee problem while Harry Winks had to be stretchered off after receiving oxygen on the side of the pitch, having appeared to suffer a serious injury following an innocuous collision in front of Burnley's dug-out.
Eric Dier was ushered into midfield alongside substitute Mousa Dembele for the second half as Pochettino returned to a 4-2-3-1 formation.
The tactical tweak almost produced an immediate impact when Vincent Janssen teed up Ben Davies only for Heaton to deny the Welshman.
Despite being forced to operate in difficult circumstances Spurs began to find their groove and Dier was the coolest man on the pitch when he maintained his composure in a crowded penalty area and slotted into the top corner.
The travelling fans erupted, all too aware of the significance of the goal, and they were celebrating again when Alli picked out substitute Son to tap home at the back post.
'Chelsea - we're coming for you' came the chant from the away end. Spurs still believe - and rightly so.