A Robin van Persie free-kick and Samir Nasri's 13th goal of the season put the visitors in control before an own goal from Roger Johnson added to City's misery.
The Gunners remain in third spot, two points adrift of United and City, after a sixth away success of the season.
Arsenal were worthy winners and carved out a host of gilt-edged opportunities. Skipper Cesc Fabregas pulled the strings in midfield and City struggled to cope with the runs of Van Persie and Nasri.
Only poor finishing and a couple of fine saves from Ben Foster kept the scoreline respectable.
Birmingham had already slipped back into the relegation zone before kick-off and failed to match the quality of their performance in the midweek draw with Manchester United.
Manager Alex McLeish will not press the panic button and is aware of the need to bring in a quality striker during the January transfer window.
But he will also be concerned at the way his normally frugal defence was cut to ribbons by the rampant Gunners.
Arsenal started brightly and Theo Walcott skated past the challenge of left-back Liam Ridgewell but shot into the side netting from a narrow angle.
Foster was called into action for the first time in clinging onto a low drive from Van Persie after being teed up by Fabregas.
Arsenal were in the ascendancy and after 13 minutes Van Persie put them ahead when Scott Dann was adjudged to have brought down Van Persie just outside the area.
Van Persie's free-kick took a deflection off Lee Bowyer and flew past Foster into the corner of the net.
Birmingham tried to retaliate and it needed a fine save from Lukasz Fabianski after Fabregas had fouled Craig Gardner.
Former Arsenal midfielder Sebastian Larsson's free-kick was lifted over the defensive wall and Fabianski had to go full stretch to his right to keep out the effort.
Arsenal were still looking the more dangerous and a reverse pass from Nasri played in Van Persie only for his control to let him down.
A defence-splitting pass from Fabregas picked out Walcott but his first touch left a lot to be desired and Foster was able to collect.
Birmingham were struggling to move out of second gear although Johan Djourou was yellow carded after hauling down Jerome.
The home side had claims for a penalty turned down when Van Persie looked to control the ball with his arm after Johnson headed back across goal.
Arsenal were carving out plenty of chances and Van Persie found himself with only Foster to beat, but he tried to clip the ball over Foster and failed to make proper contact on his shot.
The game was now more open and Johnson squandered a good opportunity when he volleyed wide after Jerome had flicked on a Larsson free-kick.
But Arsenal were soon back on the offensive at the start of the second half and the influential Fabregas, on his 200th Premier League appearance, created an opening for Walcott.
However, the England winger completely miskicked across the face of goal.
The chances continued to fall to Arsenal and Wilshere should have marked his 19th birthday with a goal. Walcott's chip was turned back to an unmarked Wilshere eight yards out but he sidefooted over the bar.
Foster then did well to block an effort from Nasri after a mistake by Johnson had allowed him a free run at goal, but it came as no surprise when Nasri made it 2-0 after 57 minutes.
He exchanged passes with Fabregas before driving a low shot past Foster just inside the box.
Eight minutes later it became 3-0 with an own goal from Johnson. Nasri set up Fabregas, whose shot was blocked by Foster, but the loose ball bounced off Dann and then Johnson and over the line.
Arsenal were now rampant and Fabregas had a free run into the area but fired high and wide. Foster was equal to a low drive from Walcott who was in acres of space and then turned a low shot round the post from substitute Andrey Arshavin.
Blues looked well beaten and Dann needed to make a last-ditch tackle to deny Van Persie a second goal.