Alexander Hleb gave Blues a 1-0 lead at Bloomfield Road on Tuesday night but former Blues striker DJ Campbell equalised for the hosts midway through the second half.
However, Scott Dann secured all three points for Blues when he netted the winner in the last minute of normal time.
"I am delighted to win the game first and foremost," McLeish said.
"I suppose people will stop asking about the away record now, although they might be asking me about it only being one (away win) in six months time.
"But it is three valuable points and that is the main thing.
"It is a great lift..
"The players themselves know they played extremely well in the first half and that they didn't do so well in the second half, but there is obviously a big psychological effect when you find yourself at the bottom of the table.
"Sometimes people can be a little bit uptight and apprehensive and that was a little bit evident in the second half.
"I felt we had enough chances in us to have had a bit of daylight before Blackpool came back into it."