After United's lack-lustre display at West Ham on Saturday, the points were more important than the performance of Sir Alex Ferguson's side.
The only goal of the game came from one of its few memorable moments. Michael Carrick's long pass found Tevez in the centre of the Birmingham half.
Tevez played the ball to Cristiano Ronaldo whose glorious back-heel sent the Argentinian racing clear into the Birmingham box. His clinical finish saw him slot the ball beyond goalkeeper Maik Taylor and into the net from 12 yards.
Tevez was United's star man and he should have had more than one goal for his brilliance.
In the sixth minute, a superb move involving Anderson and Ronaldo saw Patrice Evra released on the left side of the Birmingham box and he crossed the ball to the near post where Tevez hit the woodwork with a header from eight yards.
Tevez was denied by the woodwork again 12 minutes after the interval.
Ji-Sung Park found the Argentinian on the right side of the Birmingham penalty area.
With Blues skipper Liam Ridgewell behind him, Tevez lost his balance and fell to the ground, but was soon up and into a position where his ten-yard shot hit the post.
Ronaldo tried to tuck home the rebound as he beat two Birmingham defenders before seeing his goalbound shot blocked by Gary McSheffrey.
United had other chances to make it a more convincing scoreline only to find Taylor in outstanding form between the Birmingham posts.
The Northern Ireland international's best save came in the 51st minute. Park, returning for his first start in nine months, produced a good cross from the right byline and Ronaldo powered in with a bullet header from eight yards at the near post which Taylor kept out with his legs.
Taylor went on to keep out further efforts from Tevez and Ronaldo as United searched for a second goal.
Although United goalkeeper Tomasz Kuszczak had a quiet game, the home team had one or two worrying moments in the closing stages after Mikael Forssell had replaced Cameron Jerome.
Forssell went close with two shots on the turn, but for Birmingham, Old Trafford remains a graveyard of hope.
They last won on the ground 30 years ago to the day with their last goal at United coming in 1981.
One worrying point for United was the departure of Tevez in the 71st minute when he hobbled off with what looked like a nasty ankle injury.