Arsenal had been overwhelming favourites to end their six-year trophy drought but a mix-up between defender Laurent Koscielny and goalkeeper Wojciech Szczesny allowed Martins to secure Birmingham's first major trophy since 1963.
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"Relatively speaking, it is the greatest moment of my career, in terms of personal achievement, when you consider the team we were up against," Birmingham manager Alex McLeish told reporters.
"For a little club like Birmingham to take on the might of Arsenal, it's fantastic. We weren't given a prayer today and think even the bookies got it totally wrong.
"I said myself that Arsenal would win nine times out of 10. But we believed. It was a titanic performance."
"I am bitterly disappointed. I hope this won't have a negative affect on our season," Wenger said afterwards.
"You have to be positive because he is a young boy," he added.
"Hopefully, he will pick himself up and respond well (in the FA Cup replay against Leyton Orient) on Wednesday.
"When we conceded the goal there was no time for us to respond."
Nasri set up Andrei Arshavin for Arsenal's first real scoring chance when the stocky forward had time to turn and shoot and Ben Foster did well to save with his feet.
Birmingham, with 6ft 8ins (2.03 metres) tall Zigic as their target man, tended to go for the high ball and the forward's aerial power brought the opener after 28 minutes when he outjumped Szczesny to head his side in front.
