Blues went into the game 2-1 down after the first leg at Upton Park and when Carlton Cole gave West Ham a 1-0 lead at St Andrews they seemed destined to miss out on a Wembley appearance against Arsenal next month.
However, McLeish replaced Matt Derbyshire with Serbian striker Zigic at the start of the second half and Blues were transformed.
Goals from Lee Bowyer and Roger Johnson levelled the tie and Craig Gardner's strike in extra time gave Blues a 3-1 win on the night and a 4-3 on-aggregate success.
"We were down and out," McLeish told Sky Sports after the match.
"I said to the players at half-time, to all intents and purposes, we are out of the cup so I have got to make a change right away and see if I can make an instant impact.
"I said if you get a chance to cross the ball do it. Put it in the box with big Zigic in there.
"It was his most effective game for us and when he plays like that he can be very difficult indeed to handle."
"I was probably more relaxed than I have been in recent weeks in the Premier League. I don't know why, but I did have a calm about me.
"I just had to trust the players and the players had to trust each other. We tried our best in the first half but we huffed and puffed without really troubling them.
"Zigic made the difference.
"I will have enormous pride leading the side out at Wembley.
"I have been involved in a lot of semi-finals and have got to finals in Scotland in my career, so to come to England and get to Wembley is a dream come true."