Blues got off to a dream start when Peter Lovenkrands was given the freedom of the penalty area to head home from a Darren Ambrose free-kick after four minutes.
But Coventry were level on 19 minutes. Kevin Kilbane played a neat ball between the lines for Cody McDonald to stroke home.
Three minutes later the hosts were in front. Baker's cross went untouched and reached Kilbane, who caught the ball sweetly on the bounce to score from 18 yards.
The sides went in level at half-time after Jonathan Spector thrashed home following a one-two with Morgaro Gomis a minute before the break - but the second half failed to produce a winner.
The decisive goal arrived after seven minutes of extra-time when Kilbane's cross was not cleared, allowing Baker to smash home.
Gomis hit the bar three minutes later but that was as close as Birmingham came.
