Juan Mata and Raul Meireles scored second-half goals to take the Premier League club through to a meeting with another second-tier side, Leicester.
Roberto Di Matteo, who has stepped up from his role as Villas-Boas' assistant to lead the team for the rest of the season, left Frank Lampard, Didier Drogba, Michael Essien and Daniel Sturridge out of his starting lineup.
Although the influence of experienced players was widely cited by British newspapers as being central in Chelsea's poor performances under Villas-Boas, Di Matteo's decision may have been motivated by a desire to keep them fresh for the club's ailing challenge for Champions League qualification.
"My thoughts tonight go to Andre," Di Matteo said. "It was important tonight to go through to the next round, for our club and for our supporters.
"We have two months left to the end of the season. We have to try to achieve what we have set ourselves for this season. It's only possible if the 25 players in the squad are a little bit unselfish and work together."
Nikola Zigic troubled Chelsea's defence through an even first half, but Mata poked in a 53rd-minute opener after Birmingham failed to clear a cross by Ramires.
Ramires then set up Meireles seven minutes later for a 25-meter (yard) strike into the top corner to seal the team's first win without Villas-Boas.
"You have to move on. Today was the first step," Chelsea goalkeeper Petr Cech said. "The board and the directors took the decision, which you as a player have to respect."
Tottenham hosts third-tier club Stevenage in another replayed fifth-round match on Wednesday.