League Cup - Hargreaves scores in routine Man City win
Hargreaves signed on a free transfer during the summer window after being released by City’s rivals Manchester United, where the 30-year-old endured an injury-plagued four years.
The midfielder opened his City account with a superb 25-yard strike, before Mario Balotelli made the game safe after 38 minutes.
Birmingham have a packed fixture schedule this season thanks to their involvement in the Europa League and boss Chris Hughton also overhauled his side, making seven changes to the team that lost 4-1 at Southampton in the Championship at the weekend.
Manchester City dominated from the first whistle, moving the ball around comfortably and Balotelli saw his shot saved by Colin Doyle.
And the hosts soon went ahead when Birmingham failed to clear their lines and Hargreaves let fly with a pin-point shot that left Doyle with no chance.
Tevez then saw his header cleared off the line, before Balotelli was just inches from converting a low cross, instead accidentally catching Doyle in the face with a stray knee.
Birmingham improved after the break but, despite working the ball into good positions, were unable to capitalise with Adam Rooney the lone striker.
Birmingham’s Guirane N’Daw saw his close range shot well saved by Costel Pantilimon, while at the other end, Balotelli saw his audacious chip go just wide after spotting Doyle off his line.
And Doyle produced a superb double save to first deny Tevez before keeping out Toure’s follow-up as City looked to press for a third in the closing stages.