Middlesbrough will face Arsenal in the semi-finals after a dramatic penalty shoot-out victory at White Hart Lane.
Franck Queudrue netted the sudden-death winner after Taricco had hit the post with his penalty for Tottenham.
Earlier, Anderton fired Spurs ahead after 63 seconds as Boro conceded for the first time in more than 11 hours.
Michael Ricketts equalised four minutes from the end of normal time when he swept home a George Boateng cross.
Spurs made just one change to the side which lost 4-0 at Newcastle last weekend, with Rohan Ricketts replacing Gus Poyet, while Boro striker Gaizka Mendieta passed a fitness test on a toe injury to partner Massimo Maccarone.
Middlesbrough went into the tie with seven consecutive clean sheets behind them, but the visitors were left shell-shocked by Tottenham's whirlwind start.
Dean Richards started the move with an invitation for Stephen Carr to attack down the right flank.
Colin Cooper could only half-clear the ensuing cross and Anderton pounced to rifle his shot past goalkeeper Mark Schwarzer.
Boro bounced back well with Mendieta and Boudewijn Zenden both blazing long-range shots wide.
Tottenham thought they had done enough to win until Ricketts equalised four minutes after sweeping home a Boateng cross.
Ricketts almost headed home the winner for Boro in injury time but Keller pulled off a remarkable one-handed save to tip the ball over the bar.
Maccarone was slightly closer with a crisp volley on the turn, but Spurs threatened again as Robbie Keane struck the side netting from a tight angle.
Schwarzer also had to tip Ledley King's header over the bar when the midfielder rose to meet Anderton's corner.
Boro finished the stronger in extra time and Keller denied Stewart Downing to force the penalty shoot out.
Robbie Keane, Stephen Carr and Paul Konchesky all scored for Spurs, while Michael Ricketts, Juninho and Boudewijn Zenden were on target for Boro.
However, Poyet's effort was then saved by Schwarzer and while Danny Mills and Kanoute scored, Keller somehow managed to keep Spurs alive by denying Mendieta.
Taricco then struck the post and it was left to Queudrue to seal Boro's place in the semi-finals.