A late goal from Nikola Zigic earned Blues a 2-1 home success and a last four clash with West Ham. It was Blues' first win over their local rivals for five and a half years.
Blues even overcame having an earlier effort from Zigic ruled out for offside against Liam Ridgewell when they were leading 1-0.
McLeish said: "We earned the right to be in the semi-finals and I'm really proud of the players. It was a fantastic effort by them. I felt we deserved it.
"We could have been 2-0 up if the offside decision had gone our way. We finally got our reward in second half when they were probably the better side in terms of possession but never really hurt us.
"I felt really pleased for them, really delighted. They've met a lot of challenges - this team - over the past year and a half. This was another one and they've risen to it."