A 56th-minute opportunist goal by Dutch striker Ruud van Nistelrooy secured the victory that took United to 53 points -- three behind Arsenal and five ahead of third-placed Newcastle United, who have a game in hand.
Birmingham, who face Chelsea and Liverpool in their next two games, worked hard throughout but barely threatened to score and remain only two places and six points above the relegation zone.
Tuesday's clash was United's first game at St Andrews for 17 years and they went about their early business as if they were still trying to find their way around the famous old ground.
United had the lion's share of possession in the early stages but lacked an incisive edge.
Birmingham, who signed six new players during January's transfer window in the hope of staying up after their promotion last season, had only one real opportunity when Christophe Dugarry had a header cleared of the line by Juan Sebastian Veron after half an hour.
United went closer just before the break when in-form Paul Scholes unleashed a shot from a tight angle but saw it strike the underside of the bar and bounce to safety.
The goal came 11 minutes into the second-half after van Nistelrooy fashioned a chance out of nothing.
The big striker, who had barely touched the ball before that point, collected it on the penalty spot with his back to goal, surrounded by defenders, but somehow turned to score his 26th
"He turned very quickly, I think he caught them by surprise," manager Alex Ferguson told Sky Sports TV.
"It may have taken a deflection but its gone into the corner of the net and given us the three points we really needed."
From then on United produced a solid performance to give themselves another lift ahead of a testing two weeks during which they will play Manchester City in the league, Arsenal in the FA Cup and Juventus in the Champions League.
Ferguson said he was revelling being involved in such tussles.
"I think its great, that's why we are here," he said.
"It doesn't matter whether it's for the cup or for the league, it brings that exhilaration."
Birmingham manager Steve Bruce, who served with distinction as a player for nine years under Ferguson at Old Trafford, said there were some "plus points" to take from the game but overall was disappointed.
"The plan was to make it difficult for them and for an hour we did that," he said.
"From our point of view it was a bad goal to give away but it was typical van Nistelrooy -- in the box he comes alive and he has punished us."
