Brum came back from a goal down to win 2-1 at Pride Park.
McLeish told bcfc.com: "The supporters were rocking at the end and I'm sure they'll enjoy a beer tonight after what was a great win for us.
"With a quarter of an hour left it looked as though we were out of the cup but once again we showed that never-say-die spirit that has served us so well this season and turned it around.
"We weren't at our best today but there can be no doubting the guts we showed and a will to win that in the end got us through.
"It's great to be through and have our name in the hat for the FA Cup quarter-finals. Let's hope the draw is a little bit kinder than it has been and that we land ourselves a home tie!"