With Bolton's Muamba watching on television from his hospital bed, according to media reports, Spurs triumphed 3-1 to book an April 15 date with Chelsea at Wembley Stadium.
Everton will meet local rivals Liverpool in the other semi-final after beating Sunderland 2-0 in their quarter-final replay.
There was applause around White Hart Lane before kickoff in honour of Muamba and the medical staff who helped the stricken player 10 days ago.
Players from both teams warmed up wearing T-shirts with the slogan 'Uniting for Fabrice'. The 23-year-old Muamba remains in intensive care in the London Chest Hospital.
Spurs dominated the opening exchanges with opposition goalkeeper Adam Bogdan in inspired form until a late flurry of goal action sent the London club through.
New Zealander Ryan Nelsen rose at the back post to head home Rafael Van der Vaart's corner on 74 minutes, then Gareth Bale slotted home the second goal two minutes later from just inside the area.
Although Kevin Davies pulled one back for Bolton as the game approached injury time, substitute Louis Saha wrapped up the victory with a curling shot from 20 metres, much to the delight of Spurs manager Harry Redknapp.
"We couldn't have played any better than we did tonight," Redknapp told reporters. "I said at half-time if we keep playing the way we are the goals will come. It was never in doubt we would go on and win the game."
COOL FINISH
Everton, held to a 1-1 draw by Sunderland at Goodison Park 10 days ago, went through thanks to a cool Nikica Jelavic finish in the first half and David Vaughan's own goal on 57 minutes at the Stadium of Light.
Jelavic, bought from Scottish club Rangers in January, was praised by team-mate Tim Cahill.
"He has changed a lot of our games," the Australian midfielder told ESPN. "It doesn't matter if he misses, we care about each other a lot and we've got each others' backs.
"It's about time we produced something like this. It was all about us today."
Sunderland pressed for an equaliser but the wind went from their sails when Vaughan, who had been on the pitch for four minutes, inexplicably put the ball into his own net as a tame Jelavic shot crept towards the goal-line.
Everton and Liverpool will meet at Wembley on April 14.
Redknapp said the outcome of the cup was impossible to predict.
"You wouldn't like to pick a winner," he said. "Everton and Liverpool are both capable of winning the cup. Chelsea have gone to Benfica tonight and won in the Champions League. It is wide open really."
