The visitors made six changes from the side that won at Derby on Sunday - and it showed in a disjointed first 45 minutes in which Luton were the better team.
Bell was the main danger to Everton, blasting just wide from 25 yards, and he was then off target again with another long-range effort.
But Luton suffered a blow after only 19 minutes when veteran striker Paul Furlong limped off to be replaced by Calvin Andrew.
Everton's powerful young striker Victor Anichebe had been kept under control, but he escaped his marker Jaroslaw Fojut in first-half stoppage time and it needed a swift intervention from Chris Coyne to clear away from James McFadden.
David Moyes' side turned up the heat after the break, with Leon Osman forcing two fine saves from Luton keeper Dean Brill.
Alan Stubbs fired a free-kick narrowly wide and central defensive partner Lescott, who had an outstanding game, shot just inches wide.
Spring then scrambled Anichebe's header off the line as Luton came under pressure.
But Luton showed great strength of character and were able to turn up the heat on Everton as time ran out.
Everton sent on James Vaughan for his first appearance of the season after recovering from a dislocated shoulder, and he almost made an instant impact but headed just wide from eight yards.
Gravesen made his entrance three minutes into extra-time, and he played a key role as Everton finally broke the deadlock after 101 minutes.
The Dane's corner found its way through to Cahill at the far post, and he stretched to turn the ball high past keeper Brill.
Luton were acquitting themselves very well, and Lescott rescued Everton with a goal-line clearance from Spring with only eight minutes left.
Everton keeper Stefan Wessels then saved his side in the dying moments with a fine plunging save from Andrew when he looked poised to equalise.
Luton manager Kevin Blackwell: "It was a fantastic performance from start to finish and it's credit to the players.
"You couldn't tell who had the money and who didn't out there.
"David Moyes has had to throw his big guns on because I think he thought it wasn slipping away, and Tim Cahill has popped up and won it for him."
Everton assistant manager Alan Irvine: "Tim Cahill's got a fantastic knack of being in the right place at the right time
"When the ball finds him in those places he's composed enough to find the net and he's very important for us.
"In the first half we were very disappointed with our performance but we got the result."
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