Tottenham, with Tim Sherwood in charge for the first time, flew out of traps and went close twice through Jermain Defoe and the lively Andros Townsend.
Emmanuel Adebayor, restored to the starting line-up after being frozen out by Andre Villas-Boas, steered a header wide as Spurs continued to boss the opening period without truly testing Adrian in the visting goal.
West Ham were much more positive after the break but it was Spurs who broke the deadlock in the 66th minute when Defoe's cross to the far post was superbly volleyed home by Adebayor.
The Hammers drew level, however, with 10 minutes to go when Jarvis beat Hugo Lloris at his near post after good work by Modibo Maiga and Matt Taylor.
And Sam Allardyce's men stunned the home faithful with five minutes to go when Maiga powered a header past Lloris to book their place in the semi-finals.
With the selection of Adebayor and Defoe as two of six changes in a conventional 4-4-2 formation, Sherwood made a statement.
The former Spurs captain, wearing a tracksuit and trainers, showed more energy than Adebayor in Spurs' opening attack, imploring Aaron Lennon to sprint down the right before Defoe dragged a shot wide.
Townsend shot wide from the edge of the area before another Lennon cross caused danger in the Hammers' box and Joey O'Brien headed too close to his own post for comfort in conceding a corner.
O'Brien conceded another set-piece when Adebayor attacked the right side of the area and then the Togo striker headed a Townsend cross wide.
Defoe blasted over, much to Sherwood's frustration, as West Ham struggled to break free of their own half.
Carlton Cole shot straight at Hugo Lloris as the visitors registered an effort on target for the first time after 36 minutes.
The Hammers resumed on the defensive and registered a second shot on target in strange circumstances. James Collins kicked the ball back to Spurs following an injury to George McCartney, but instead of a tame pass to a member of the opposition, the Wales defender booted the ball from inside his own half and forced Lloris to back-pedal to save.
Townsend fed Defoe, who ran down the left before crossing to the back post where Adebayor did the rest, converting with a thunderous right-footed volley which went in off the underside of the bar. Adebayor has now scored three times in his last four Spurs starts.
Mohamed Diame, on for Joe Cole, forced Lloris to save low to his left as the Hammers sought a swift response.
Townsend pulled up worryingly for Spurs and departed with 17 minutes remaining, with Nacer Chadli on in his place.
Lloris saved a Matt Taylor effort from 25 yards before Adebayor made way with 12 minutes remaining to a rousing reception from the home faithful.
But instead of Tottenham closing out victory, West Ham equalised when Maiga flicked on goalkeeper Adrian's long ball to Taylor. With his back to goal, Taylor turned and fed Jarvis on the left and the winger was granted enough space by Kyle Walker to fire in.
O'Brien made a fine challenge to deny Chadli after the midfielder broke into the West Ham box and shaped to shoot.
The Hammers then struck again after substitute Ravel Morrison found Diame.
Diame's deep cross was met at the far post by Maiga, whose header found the corner for his first goal in a year.
Sigurdsson was denied by Adrian in the first of five minutes of added time as the visiting fans taunted Tottenham and Sherwood, exclaiming "You're getting sacked in the morning".
Maiga then struck the crossbar with a powerful drive and Jack Collison forced Lloris to save as West Ham finished the contest on top as they sealed their progress.