Conor Gallagher scored the game’s only goal when he drove through a crowded box early in the first half.
Kyle Bartley went close to adding a second for the hosts after the break but his header was superbly kept out by Blades goalkeeper Aaron Ramsdale.
Baggies keeper Sam Johnstone twice prevented the visitors from equalising with excellent saves, keeping out ex-West Brom winger Oliver Burke’s powerful header before somehow stopping Oli McBurnie‘s fierce drive.
George Baldock also had a golden opportunity to equalise but somehow bent a shot over the bar when unmarked inside the box.
The visitors threw players forward in the closing stages in a desperate attempt to rescue a point but when Lys Mousset lashed over from six yards out deep into the seven minutes of time added on, it was clear it was not gong to be the Blades’ night.