Both teams paid the price for wholesale changes from their Premier League matches at the weekend, with City losing 4-2 after extra time at home to a spirited Villa, and the Toffees beaten 2-1 by Championship Leeds at Elland Road.
Premier League champions City took the lead when Mario Balotelli turned and scored on 27 minutes, but a Gareth Barry own-goal just before the hour mark put his former side level.
Aleksandr Kolarov put City back in front with a trademark free-kick, but Gabby Agbonlahor levelled with 20 minutes remaining in a frantic 11-minute period of the second half.
Carlos Tevez hit the bar for City as extra time beckoned, during which Villa pulled clear thanks to a Charles N'Zogbia tap-in following a shot by Agbonlahor, who added a late fourth with a deflected effort.
Everton, third in the Premier League, fell behind after four minutes when Aidy White jinked his way through the heart of the visitors' defence and curled the ball into the top corner.
Leeds doubled their advantage when a Danny Pugh shot was prodded into the bottom corner by Rodolph Austin in the second half before Sylvain Distin headed in a late consolation.
Everton piled on the pressure at the death and seemed to be denied a good shout for a penalty when Victor Anichebe was brought down in the box, but Leeds held on for a famous win.
Victor Moses scored his first goal for Chelsea as they emphatically booked their place in the fourth round with a 6-0 win over Wolverhampton Wanderers.
Goals from Gary Cahill, Ryan Bertrand, Juan Mata, Oriol Romeu, Fernando Torres and a first for Moses since his move from Wigan gave the hosts – captained by John Terry just hours after the second day of his FA hearing on a race charge – an easy win that could have been bigger.
Two moments of individual magic were enough for 10-man Sunderland to see off a spirited MK Dons 2-0 at Stadium:mk. Lee Cattermole had been sent off for a reckless tackle on Adam Chicksen on 32 minutes, but a superb free-kick from Craig Gardner put the visitors ahead, while James McClean added a fine second.
Garry Monk scored in the 90th minute for Swansea City who beat third tier Crawley 3-2, while Wigan Athletic won 4-1 in an all-Premier League encounter against West Ham United.
Other matches saw Middlesbrough win 3-1 at League One Preston, while Southampton beat Sheffield Wednesday 2-0 thanks to a brace from Jay Rodriguez.
Swindon claimed another Championship scalp with a 3-1 win at home to Burnley, while Burton Albion threw away a two-goal lead at Bradford, losing 3-2 after extra time.