Neville sent goalkeeper Petr Cech the wrong way after Chelsea full-back Ashley Cole blasted his penalty high and wide as Everton ended the Londoners' dreams of becoming the first club for 125 years to win the FA Cup three times in succession.
The fairytale run of minnows Crawley Town also ended although the non-League side left Old Trafford with their heads held high after only losing 1-0 to Premier League leaders Manchester United in their fifth round tie.
Crawley, bidding to become the first team from outside the League to reach the last eight for 97 years, fell behind to a 28th minute header by Wes Brown but almost equalised in stoppage time when Richard Brodie headed against the top of the bar.
Premier League clubs Birmingham City and Stoke City beat lower league opposition far more easily than United.
Birmingham, already through to the League Cup final against Arsenal later this month, reached the quarter-finals with a 3-0 win over League One Sheffield Wednesday with goals from Jean Beausejour, Obafemi Martins and David Murphy.
Stoke beat League One leaders Brighton by the same score with goals from John Carew, Jon Walters and Ryan Shawcross.
After the win over Chelsea, Everton captain Neville told the BBC: "I knew I was going to score, I knew where the ball was going to go - and that's half the battle.
"We've been practising penalties all week. The manager just told us to hit them as hard as possible.
"My biggest worry was that there was a big divot in the penalty spot. Petr Cech looks really big but luckily he dived the wrong way."
After a goalless stalemate in 90 minutes of end-to-end football, Chelsea took the lead in the last minute of the first period of extra-time when midfielder Frank Lampard scored with a low volley from eight metres.
NO TORRES
With a minute of extra time remaining Leighton Baines equalised with a left-footed free-kick to force the match into a shootout.
Everton trailed 2-0 on penalties after Lampard and Didier Drogba converted their spot-kicks and Baines saw Cech save his penalty.
Phil Jagielka scored to pull Everton back to 2-1 before Chelsea's Nicolas Anelka, taking virtually no run up, had his spot-kick saved by Everton goalkeeper Tim Howard and Mikel Arteta scored to make it 2-2.
Michael Essien and Johnny Heitinga made it 3-3 before Cole missed and Neville scored to give Everton their stunning victory and end Chelsea's hopes of emulating Blackburn Rovers, the last team to win the Cup three years in a row in 1884, 1885 and 1886.
Chelsea, playing without their recent high-profile signings Fernando Torres and David Luiz who were ineligible, started lethargically but created more chances and finally went ahead after 104 minutes when Lampard lashed in from close range.
Everton thought they had won the match in the last minute of normal time when Cech could only parry a long-range
