Billy Sharp scored in both halves as the Blades moved to within a point of the play-off places, with two games in hand on sixth-placed Nottingham Forest, in a ridiculously tight promotion battle in the Championship.
Albion showed signs of getting back into the game after Sharp’s deflected opener, before Livermore was shown a straight red for a poor challenge on Conor Hourihane.
“Up until the red card I thought we were the better team,” Bruce said after the game.
“I was pleased with the way we started, but I thought the red card was harsh. It was a decent challenge.
“There was a tackle in front of me not long earlier and the referee didn’t even give a free-kick, never mind a booking.
“It’s that inconsistency and I thought the referee got it wrong.”
Defeat continues the Baggies’ poor recent run of form, but Bruce says the Blades can be an example to his side after their own resurgence under Paul Heckingbottom – who has won seven of his nine league games since taking charge.
“We huffed and puffed,” Bruce added.
“You have to look at Sheffield, they’re marching now but two months ago they decided to make a change. There’s still a lot to play for and a long way to go.
“We asked the players to change in six days what they’ve done in six months but they’re suited to it, so let’s see if it works.”