Premier League strugglers Burnley were fast out of the blocks and raced into a 2-0 lead after eight minutes in the third-round replay at White Hart Lane, with the original encounter having ended 1-1 at Turf Moor last week.
Marvin Sordell scored his first Burnley goal since joining in the close-season with a stunning half-volley before Ross Wallace's free-kick went in via a fortuitous deflection off Roberto Soldado.
The hosts quickly responded with a well-taken goal from Paulinho that rounded off an exhilarating opening 10 minutes and Etienne Capoue drilled home the equaliser on the stroke of half-time.
After an opening 45 minutes that had all the hallmarks of an FA Cup classic, the second half went to script as Vlad Chiriches took advantage of an error from Burnley goalkeeper Tom Heaton to head Spurs in front.
Mauricio Pochettino's side showed their superiority when Danny Rose added a fourth in the 52nd minute and that proved enough to clinch a round-four showdown at home to Leicester City.
It took just three minutes for the opening goal to arrive as Sordell opened his Burnley account in style.
Kieran Trippier played a long ball down the centre of the pitch and Sordell let it bounce before unleashing an unstoppable half-volley past Michel Vorm into the left-hand corner from the edge of the box.
Burnley left the home fans stunned by doubling their advantage five minutes later when Wallace's 25-yard free-kick struck Soldado and wrong-footed a helpless Vorm.
The fast-paced start continued in the 10th minute when Spurs halved the deficit.
Andros Townsend's cross from the right picked out Soldado and his flicked header found Paulinho, who controlled the ball on his chest before swivelling on the spot and calmly volleying home from close range.
Townsend then teed up Soldado for what appeared a simple chance, but the Spaniard inexplicably lifted his shot onto the crossbar from five yards.
The home side were level in first-half injury-time, though, as Ben Davies' cross was only half-cleared and Capoue lashed home the equaliser from the edge of the area.
It took just four second-half minutes for Tottenham to complete the turnaround.
Heaton flapped at Paulinho's inswinging corner from the right and Chiriches took advantage to nod home at the back post.
Spurs made it 4-2 in the 52nd minute when Benjamin Stambouli's excellent cross-field pass found Soldado and his low centre from the right was met by the on-rushing Rose.
Pochettino's men were well in the ascendancy and Nacer Chadli flashed a cross along the six-yard box, before Soldado dragged wide from distance with just over 20 minutes remaining.
With the tie all but over, Pochettino introduced forward Joshua Onomah for his Tottenham debut.
The 17-year-old had little opportunity to make an impact, though, as the hosts comfortably wound down the clock to see off Sean Dyche's men.