Birmingham boosted their promotion prospects after edging past Derby with a second-half header from Lee Carsley at St Andrew's.Derby dominated the first half, with Kris Commons blasting over and Rob Hulse denied by goalkeeper Maik Taylor.The hosts picked up the pace after the break and Hameur Bouazza crossed in from the right for Carsley to nod in.Derby piled forward with Taylor tipping over Hulse's header late on and Carsley clearing Miles Addsion's effort away.Birmingham manager Alex McLeish:"It was hugely important just to win. We would love to come back after a dismal display and play like Brazil, but it's never going to happen."You could see nerves, but the spirit was there and the players showed energy and urgency in every part of the game."We needed that confidence lift, the players are delighted with the win."Derby boss Nigel Clough:"I'm very, very pleased with the performance and with everything but the result."We kept the ball well, created chances and a couple of the lads are saying it's the best we've played all season."We're in the bottom three but there were a few surprising results tonight. We must take positives from the game."Birmingham: Maik Taylor, Larsson, Martin Taylor, Ridgewell, Murphy, Bouazza, Carsley, Bowyer, Fahey, McFadden (Jerome 68), Phillips (Sinclair 76).Subs Not Used: Doyle, Queudrue, Johnson.Booked: Ridgewell, Bowyer.Goals: Carsley 59.Derby: Bywater, Connolly, Albrechtsen, Addison, Stewart, Barazite (Sterjovski 73), Green, Savage, Teale (Villa 63), Commons, Hulse.Subs Not Used: Carroll, Barnes, Nyatanga.Booked: Connolly.Att: 15,330Ref: Graham Salisbury (Lancashire).