Burnley raced into a 2-0 lead after just seven minutes of the third-round replay, Marvin Sordell breaking a 12-game duck and Ross Wallace’s free-kick deflecting off Roberto Soldado.
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Paulinho hit back for Spurs in the tenth 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 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 Michel 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 squirmed into the net.
The breakneck opening took another twist when Paulinho dragged Spurs back into the contest in the tenth 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 Brazil midfielder obliged with a spin and finish.
Tottenham’s frustrations peaked when Soldado hit the bar from six yards. Townsend’s low cross had Burnley goalkeeper Tom Heaton beaten bar the tap-in, only for Soldado to miscue horribly.
But, just when Spurs feared no more reward in the half, 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, the defender making no mistake at the far post as Heaton flapped at Townsend’s corner.
Rose then cemented Spurs’ advantage by sliding Soldado’s pass home from close-range.
Southampton never looked in any real danger of losing their third-round replay at Ipswich as Ronald Koeman’s side claimed a 1-0 win.
Saints arrived in Suffolk looking to avoid another cup exit to lower-level opposition. The Premier League high flyers were last month embarrassed in the Capital Cup quarter-finals by third-tier Sheffield United, but they did not slip up against the Championship promotion chasers as Shane Long’s first-half strike saw them through.
Saints were pragmatic rather than pretty, with the decisive goal coming after 19 minutes when Long reacted quickly to produce a lovely finish after James Ward-Prowse was tackled.
Bradford booked a coveted trip to Chelsea by sending ten-man Championship strugglers Millwall spinning to a 4-0 defeat at Valley Parade.
The Londoners never recovered from the early dismissal of defender Mark Beevers. Ian Holloway’s side were reduced to ten men in the sixth minute, found themselves 3-0 down at half-time following goals from James Hanson, Jon Stead and Andy Halliday and conceded a fourth when Billy Knott struck early in the second period.