The draw meant the Town's winning run on the road came to an end, after three successive victories at Swansea, Cardiff and Hull leading into the trip to the West Midlands.
But the point made it just one defeat in the last nine matches and maintained the ninth place Town occupied at kick-off, three points off the play-offs, with a third clean sheet in four matches.
Manager Nathan Jones made one change from the midweek draw with Huddersfield, captain Sonny Bradley returning to the starting line-up for the first time since the 2-2 draw with Coventry City last month, replacing Fred Onyedinma who was named on the bench.
Luke Berry and Luke Freeman were back in the matchday squad after recovering from injury, replacing youngsters Casey Pettit and Zack Nelson on the substitutes' bench.
The hosts started brightly, Brandon Thomas-Asante hitting the side-netting from close range inside the opening minute before Martin Kelly headed wide at the far post from the game's first corner soon after.
Ethan Horvath had to dive low to his left to keep out a Jed Wallace effort after quarter-of-an-hour, but the Hatters began to make inroads, James Bree twice sending in dangerous crosses from the right, the second leading to Henri Lansbury driving in a corner that Dan Potts headed at goal, only for Okay Yokuslu to clear off the line.
Thomas-Asante hit the post with a low shot on the turn as the Hatters managed to clear a 25th-minute corner only as far as the edge of the box, and Town were fortunate not to be behind.
The Championship's player of the month for September, Carlton Morris, had his first sight of goal in the 33rd minute when strike partner Adebayo slipped him in on the right side of the penalty area, but his shot found the outside of the side netting.
The scoreline remained blank heading into the break, and it was the Baggies who came out of the blocks quickest in the second half too, Thomas-Asante setting Conor Townsend up for a left-footed shot that Horvath tipped round his near post.
Dara O'Shea headed a Grady Diangana corner onto the roof of the net, and the Hatters then threatened briefly, but Clark couldn't connect cleanly when attempting to volley Bree's cross goalward.
Bradley became the first player to be shown a yellow card for a 57th-minute foul on Thomas-Asante, then Yokuslu soon joined him for a late challenge on Morris.
Town came closest when Clark fed Adebayo in the box on 67 minutes, and the striker's shot deflected up off O'Shea's leg, forcing a backpedalling Alex Palmer - one of Town's emergency loan goalkeepers last season - to tip over his bar.
Jones made his first change with just over 20 minutes remaining, introducing Harry Cornick in place of Morris, but it was Adebayo who had the Town's next effort on goal, his own flick-on rolling back into his path from Erik Pieters' poor touch, but the 25-yard side-footed shot was comfortable for Palmer to gather.
The keeper was in the right place to hold onto Potts' left-footed volley from Cornick's long throw on 79 minutes, and the Hatters could not find a way through in the closing stages, while remaining comfortable at the other end as home fans vented their anger towards their own manager, Steve Bruce.
Attendance: 21,550 (1,385 Hatters)
