The visitors had dominated possession throughout, although oddly it was Ipswich who had the better chances and both Jaime Peters and Tamas Priskin missed one-on-ones to clinch the points.
The Baggies were the first to threaten in the tenth minute when Graham Dorrans swung in a corner which an unmarked Jonas Olsson glanced across the face of goal to Roman Bednar who skied over from an angle.
Bednar's angled shot was then tipped away by Arran Lee-Barrett in the Town goal before Jon Stead and David Norris sent efforts over the top for the hosts.
Lee-Barrett saved Brunt's 20-yard drive after a neat passing move before the keeper was cautioned after racing out of his box to clatter Bednar as he moved onto a through ball.
David Norris shot across the face of goal from Stead's cross but it was the visitors who were carrying the greater threat and Bednar's glancing header forced a diving save from Lee-Barrett and Gareth McAuley did well to block Robert Koren's goalbound drive.
Ipswich started to come into the game more as half-time approached and Pablo Counago found Norris whose low drive forced an excellent diving save from England keeper Scott Carson, with McAuley powering a free header wide from Leadbitter's ensuing corner.
McAuley cleared off his own line to deny Brunt after Lee-Barrett had missed a Dorrans cross, but Town had the best chance of the half when Counago put Peters in but he tamely shot straight at a grateful Carson.
West Brom made the better start to the second half, with Bednar glancing one header narrowly wide from a Marek Cech cross and helping on a Jerome Thomas cross which David Wright cleared from the lurking Brunt.
Against the run of play Olsson was harshly penalised for a foul on Damien Delaney as they contested sub Carlos Edwards' long throw and Leadbitter sent Carson the wrong way from the spot.
Carson denied Leadbitter's firm drive while West Brom threw on Ishmael Miller and Simon Cox to go three up front and it was Miller who had a chance when the ball pinballed to him in the box, but his shot was tame and easy for Lee-Barrett to save.
Sub Priskin sent Edwards racing towards goal but he blazed well over before Edwards sent Priskin clear, he rounded Gabriel Tamas but his shot was parried by Carson onto the head of Jack Colback who could only steer the rebound wide.
Although Town had restricted West Brom to mainly long shots, the pressure was building and in the third of four added minutes Gonzalo Jara's cross was only cleared as far as Cox whose deflected shot fortuitously bounced perfectly into the path of Brunt to net the late leveller.