Mauricio Pochettino's side were rocked as Marvin Sordell and Ross Wallace struck for Burnley in the first eight minutes of the third round replay at White Hart Lane.
But Brazilian Paulinho gave Spurs a lifeline soon after Wallace got Burnley's second and French midfielder Etienne Capoue equalised before half-time.
That set the stage for a blistering start to the second half, with Romanian defender Vlad Chiriches and left-back Danny Rose scoring in the space of three minutes to ensure Spurs earned a home tie against Leicester in the fourth round later in January.
It was exactly the kind of spirited performance Pochettino needed after his side crashed to a surprise 2-1 defeat at struggling Crystal Palace in the Premier League on Saturday.
Pochettino had made seven changes from the Selhurst Park meltdown, with in-form striker Harry Kane left on the bench, while Burnley boss Sean Dyche, whose priority is avoiding relegation from the top-flight made five changes from the side that beat QPR at the weekend.
All the Tottenham alterations seemed to unsettle the hosts during a chaotic opening which saw the Clarets take the lead in the third minute.
Tottenham defender Jan Vertonghen was caught out of position by Kieran Trippier's hopeful long ball and Sordell took advantage to race clear to slam a left-foot half-volley past Michel Vorm from just inside the penalty area.
It was Sordell's first goal for Burnley since his move from Bolton in the close-season and there was even better to come for Burnley as the visitors doubled their lead just five minutes later.
Pochettino's men had only themselves to blame for sloppily conceding the first, but Burnley's second goal was a simple case of bad luck as Wallace's free-kick took a huge deflection off Roberto Soldado in the Spurs wall and flashed past the wrong-footed Vorm.
Yet, rather than crumble after such a shocking start, Spurs hit back decisively and reduced the deficit when Andros Townsend's cross was headed on by Soldado to Paulinho, who controlled on his chest before firing home from close-range in the 10th minute.
- Miss of the season -
Spurs pressed for a first half equaliser, but Soldado's dismal time in north London continued as the Spanish striker produced a candidate for miss of the season.
A rapid Tottenham break ended with Townsend sliding over a cross that left Soldado with just Burnley goalkeeper Tom Heaton to beat, but the Spaniard somehow managed to hit the bar from no more than six yards.
Fortunately for Soldado, Spurs spared his blushes in the first half stoppage-time when Capoue latched onto Ben Davies' cross and drilled a superb strike past Heaton from 20 yards.
Spurred on by that well-timed leveller, Pochettino's side took the lead four minutes into the second half when Heaton missed Paulinho's corner, leaving the unmarked Chiriches with the simple task of bundling home on the goalline.
And the tie was effectively over three minutes later when Rose slid in to convert Soldado's low cross from close-range.
Source : AFP
Source: AFP