Brunt was recalled by caretaker manager Jimmy Shan to make his first start in 11 games and gave Albion a 19th-minute lead with a free-kick.
Swansea were then handed the chance to equalise from the penalty spot five minutes before half-time when City skipper Mike Van Der Hoorn was fouled by Craig Dawson.
ButSwans striker Bersant Celina lost his footing taking the penalty and could only mis-kick the ball two yards to his right.
Swansea slipped further behind when Mason Holgate scored in the 54th minute before Jay Rodriguez added a third five minutes from time as City boss Graham Potter, who played for Albion between 1997 and 2000, watched his side crash to a fifth successive away defeat.
In their first game since the dismissal of manger Darren Moore, Albion were lucky not to fall behind after 13 minutes when the post came to their rescue.
Matt Grimes fed a ball down the left wing to Wayne Routledge whose cross flew across the face of the Albion area to Connor Roberts.
His angled shot beat goalkeeper Sam Johnstone but cannoned to safety off the far post.
Albion then cashed in fully on that let-off when Brunt gave them the lead.
Swansea defender Kyle Naughton was penalised on the right wing after a crude challenge on Holgate and Brunt scored direct from the free-kick with a low cross that went through a crowded goalmouth and into the far corner of the net.
Albion almost doubled their advantage four minutes later from another Brunt free-kick but Ahmed Hegazi was inches wide with his shot.
Albion were then grateful to Johnstone for keeping their lead intact as the former Manchester United goalkeeper pulled off two stunning saves in the space of 60 seconds.
First he stopped a point-blank shot from Routledge in the 28th minute after Nathan Dyer had an effort blocked and then followed that up with a one-handed stop to deny Daniel James from close range.
Johnstone then tipped a drive from Celina over the bar after 38 minutes before the Kosovo international forward endured his penalty horror.
Albion doubled their advantage when on-loan Everton full-back Holgate scored his first goal since his arrival from Goodison Park when he headed home Brunt's right-wing corner.
Swansea's fate was then sealed by Rodriguez who pounced from 10 yards after latching on to a mis-placed clearance from Cameron Carter-Vickers.