Mata and Meireles give Di Matteo winning start
Roberto Di Matteo, appointed interim coach for the rest of the season after Villas-Boas was sacked following a sequence of one win in seven matches, was all smiles after starting his reign with a victory.
The visiting Chelsea fans began the match at St Andrews with incessant chants for their former manager Jose Mourinho, who led the club to two Premier League titles in 2005 and 2006, but ended the tie with loud cries of 'There's only one Di Matteo'.
"Mourinho has a big legacy at this club but I have a connection with the supporters because I've also played for them," the former midfielder told ITV Sport.
"I think our attitude was right tonight, the commitment was right, the passion was there and we had a clean sheet. But the most important thing was to get through to the next round and to show that we cared.
"Everybody was involved tonight, the people on the bench too. That's the only way we are going to reach our targets this season," added Di Matteo after Chelsea secured a home quarter-final against second-tier Leicester City on March 18.
The Italian made several changes to the team, leaving out Didier Drogba, Frank Lampard, Michael Essien and Daniel Sturridge and recalling Meireles, Fernando Torres, Salomon Kalou and John Obi Mikel.
Ashley Cole also missed out with an ankle injury and there was no discernible change from the Villas-Boas era in the first half as six-times FA Cup winners Chelsea struggled to impose themselves on Chris Hughton's Birmingham.
The visitors perked up in the second half and broke through in the 54th minute when Mata bundled the ball in from close range following a cross by Brazilian Ramires, keeping up his record of scoring in every round of this year's competition.
Meireles made the game safe six minutes later with a venomous right-foot shot from 20 metres after excellent work down the right by Branislav Ivanovic.
Spanish playmaker Mata then had a penalty saved by goalkeeper Colin Doyle, just as he did in the first game at Stamford Bridge.
Torres missed a good chance in the first half and worked hard throughout for little reward.
The Spain striker set up Sturridge with a pinpoint cross in the 76th minute and the substitute missed his kick from six metres in front of a gaping net.
Birmingham's best chance fell to Jordon Mutch in the 82nd minute but he blazed the ball wildly over.
The next task for Di Matteo is to lift Chelsea up from fifth in the Premier League, starting with Saturday's home game against Stoke City, and to overturn a 3-1 first-leg deficit in next week's Champions League first knockout round tie with Napoli.
Reports swirled in the media during the last few weeks under Villas-Boas of a rift between the Portuguese and his players but Di Matteo said it was important for everyone to pull together.
"We have two months left to the end of the season and we are all sitting in the same boat," said the Italian. "We have to help each other and try to achieve what we have set out for ourselves.
"All 25 players in the squad are going to have to be a little unselfish and work together, otherwise it's not going to be possible."