Burnley raced into a 2-0 lead in seven minutes, Marvin Sordell breaking a 12-game duck and Ross Wallace’s free-kick deflecting heavily off Roberto Soldado.
Paulinho hit back for Spurs in the 10th minute, before Etienne Capoue levelled on the stroke of half-time.
Vlad Chiriches and Danny Rose then struck after the break to secure Spurs a fourth-round home tie with Leicester on Saturday week.
Sordell’s opener caught Tottenham cold, most notably Jan Vertonghen, as a darting long-ball bisected the Spurs central defence.
The former Bolton striker sauntered through and flicked an arcing volley past the helpless Michael Vorm to hand Burnley the third-minute lead.
Just four minutes later Burnley were two to the good, with Tottenham again the architects of their undoing.
Wallace’s flimsy free-kick ricocheted off Soldado’s head, wrong-footed the once-more hapless Vorm and squirming into the net.
The breakneck opening took another twist when Paulinho dragged Spurs back into the contest in the 10th minute.
Soldado was able to atone for his unwitting assist for Burnley’s second goal by laying on Spurs’ first.
The former Valencia hitman flicked Andros Townsend’s cross perfectly into Paulinho’s path, and the 32-cap Brazil midfielder obliged with a spin and finish.
Burnley captain Michael Duff had to produce a sliding clearance to deny the lurking Soldado the chance to equalise, before Benjamin Stambouli fired over from 20 yards.
Tottenham’s frustrations peaked when Soldado hit the bar from six yards.
Just when Spurs feared no more reward in the half however, up popped Capoue to level in time added on. The France midfielder drove home from 18 yards after Paulinho had failed to connect with Ben Davies’ cross.
Chiriches headed Spurs into the lead just four minutes into the second-half, before Rose then cemented Spurs’ advantage by sliding Soldado’s pass home from close-range.