City lost 4-2 after extra-time to top flight rivals Aston Villa with a completely different team to the one that started the 1-1 draw against Arsenal on Sunday, while Everton made six changes and were beaten 2-1 by second tier Leeds United.
Chelsea also rung the changes but, captained by John Terry despite this week's FA hearing over charges of racial abuse, they emphatically booked their place in the fourth round with a 6-0 win over Wolverhampton Wanderers at Stamford Bridge.
City's second-string team still included multi-million pound signings Carlos Tevez, Jack Rodwell, James Milner, Gareth Barry and Mario Balotelli.
The latter drilled home a low shot to give City the lead with his first club goal since March, before Villa levelled when their former midfielder Barry deflected the ball into his own net.
Aleksandar Kolarov whipped in a free-kick to give City the advantage for the second time on 64 minutes but Gabriel Agbonlahor levelled five minutes later to force extra time.
Charles N'Zogbia pounced on a loose ball to make it 3-2 and Agbonlahor wrapped up the tie with the aid of a big deflection that caused his shot to loop over City keeper Costel Pantilimon.
"I'm disappointed for the result, Aston Villa are a good team but we didn't play well," City manager Mancini said.
"Our play is not quick, we want to take one touch too many, this is not good."
At Leeds, Everton manager David Moyes retained only Victor Anichebe, Seamus Coleman, Marouane Fellaini, Johnny Heitinga and Kevin Mirallas from the team that started the 3-0 win over Swansea City on Saturday.
The visitors had made a barnstorming start to the Premier League campaign, sitting third after five matches, but their new-look XI were quickly under pressure at Elland Road.
They fell behind when Aidy White jinked his way through the heart of the visitors' defence and curled a shot into the top corner after four minutes.
Leeds doubled their lead 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.
"I didn't think there was a weakness tonight, I thought we deserved it and make no mistake they wanted this tonight, they wanted to win," Leeds manager Neil Warnock said.
"We put our head and bodies on the line but you have to do that in the cup."
Chelsea's Terry returned to their line-up against Wolves, having started on the bench for Saturday's Premier League clash against Stoke City, seemingly able to put allegations that he racially abused Queens Park Rangers defender Anton Ferdinand out of his mind.
But it was his central defensive partner, Gary Cahill, who headed in a Juan Mata free-kick after four minutes before Ryan Bertrand stroked home from 10 me
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