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Nikola Zigic had the ball in the net once more in the 36th minute, but the linesman had his flag up for offside.
However, more great wing play by Redmond nearly doubled Birmingham's lead late in the half, as he drifted inside from the right before seeing his effort saved by Tom Heaton.
The Robins trailed at half time but threatened to find an equaliser just after the break, with Davies once again finding Baldock with just the keeper to beat. However, Jack Butland stood up well to make the save and keep his side in front.
Baldock thought that he was through on goal once more in the 54th minute, only to be denied by a fantastic recovering tackle by Paul Robinson.
Davies was proving to be a real handful for the Blues' defence, as he knocked the ball down for Baldock once more, but again the striker could not find the finish.
Birmingham went on to see out the victory, which lifts them above Ipswich Town in the table.