Premier League leaders City won at Arsenal for the first in 36 years with a late goal by Sergio Aguero earning a 1-0 victory against a much-changed home side in front of a 60,000 crowd.
Maxi Rodriguez and Martin Kelly scored in the second half as Liverpool beat struggling Chelsea 2-0 - their second win at Stamford Bridge in 10 days - to heap more pressure on the West London side's Portuguese manager Andre Villas-Boas.
Blackburn Rovers' dismal season took another turn for the worse when the Premier League's bottom club lost 2-0 at Championship side Cardiff City, who have reached the semi-finals for the first time since 1966.
There was also a minute's silence at Emirates Stadium for Speed with spontaneous applause breaking out for the former Leeds United, Everton, Newcastle United and Bolton Wanderers midfielder who was found hanged at his home in Cheshire.
"It was a difficult win and Arsenal played very well," Mancini said on Sky Sports. "It was a great goal and counter-attack is the perfect goal. It's important we won and we want to try and go to the final."
Arsenal manager Arsene Wenger felt his side had deserved more. "We feel harshly done by but it's part of the learning process," he said. "We were closer to winning the game than losing it. We were caught on the counter-attack for the goal and we were naive. Overall the young players have done well."
City last won the League Cup in 1976, whereas Liverpool are the competition's most successful club with seven triumphs, albeit none since 2003.
Liverpool were worthy winners at Chelsea, dominating the first half in which Andy Carroll had an early penalty saved by Chelsea's reserve keeper Ross Turnbull.
Liverpool, who enjoyed a 2-1 Premier League victory at Chelsea earlier this month, doubled their lead shortly after when Bellamy's free-kick was headed in by Kelly.
