Kevin Phillips made the most of being handed his first league start of the season by opening the scoring in the fourth minute and Craig Gardner made it 2-0 just before the hour mark before Johan Elmander pulled one back for Wanderers.
"They got a wee scent of blood there and they put us under the cosh, but the guys stood up to it magnificently at the back again," said McLeish.
"It was a game we felt we really had to win. I don't like to use that term `must-win' because what if you don't? Do you throw the towel in?
"But it was a big, big result for us.
"When everyone knows that to win will be a big, big result, it heaps the pressure on, but they have responded magnificently."