A late Franck Queudrue header earned Blues a 1-0 victory in a game they almost completely dominated from start to finish.
"It would have been a bit of a travesty if we hadn't of won that game," McLeish said after the match.
"It's as much as we've dominated a game the entire season, but saying that, Bristol are a very slick side and they counter-attacked well.
"I thought we thoroughly deserved to win. We retained our belief right until the death, had plenty of corners, and it was a relief to see Franck get on the end of one, and then there was no stopping it.
"It was a great approach from the players and a great spirit, and I think that's why the fans were so appreciative."
