Nikola Zigic had headed Blues in front 28 minutes into the game at Wembley on Sunday afternoon, Robin van Persie equalised for the Gunners with a fine volley just before the half time break but substitute Obafemi Martins scored a last-gasp winner to earn Blues a shock 2-1 success and a place in next season's Europa League.
"I am so pleased for the fans, they have craved victory for a long time," Blues boss McLeish told BBC Sport.
"With Rangers I was expected to win trophies, even though it is difficult against Celtic. To come to England and for a club like Birmingham to beat the might of Arsenal is a dream come true.
"It is especially so for the fans who have had a long wait, those long-suffering Blues fans.
"No-one gave us a prayer, we were underdogs but sometimes the bookies don't get it right.
"After the goal I really did have the vision of the cup in my hands."