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.
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.
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.
"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."
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.
"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."
"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."
