Alexander Hleb pounced on a mistake by Stephen Crainey to put Blues ahead in the first half before DJ Campbell's 68th-minute equaliser.
The visitors eventually emerged triumphant from a pulsating encounter after Roger Johnson headed down substitute Jean Beausejour's cross in the box with a minute of normal time remaining and Scott Dann fired home to make it 2-1.
The result lifted Birmingham out of the Barclays Premier League relegation zone and up to 15th - something McLeish was far more concerned about than the fact that they had not won on the road in their previous 14 attempts.
"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."
McLeish felt his players did well to overcome the psychological burden of knowing that they were mired in the bottom three going into the game.
"It is a great lift," McLeish said.
"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 - although I felt we had enough chances in us to have had a bit of daylight before Blackpool came back into it."