A Kal Naismith own goal in the fifth minute and a Callum Robinson header in the 36th minute gave the home side a comfortable half-time lead, which they built on in the 66th minute when Karlan Grant fired home.
There was a tense finish, though, when Harry Cornick converted in the 70th minute and then, eight minutes into 16 added on - because of a serious injury to Jordan Clark following a collision with goalkeeper Sam Johnstone - Pelly-Ruddock Mpanzu did likewise, but the home side were good value for their victory.
They opened the scoring with a little help from Luton in the fifth minute. Darnell Furlong's long throw-in from the right-hand side was nodded past the helpless Simon Sluga by his team-mate, Hatters captain Naismith.
Luton struggled to apply any serious pressure on their hosts in the first period, and they duly fell further behind in the 36th minute.
An in-swinging Alex Mowatt corner kick from the right was pushed by Sluga straight to Robinson, who was waiting at the back post and simply nodded the ball in from close range - that was his second goal in as many games this season.
There was a lengthy wait, of around 10 minutes, at the start of the second half as Luton midfielder Jordan Clark received extensive medical treatment following a collision with Johnstone. The Albion keeper came to punch a long ball away and collided with the unfortunate Clark, who was tended to on the field for a long time.
Eventually he was carried off on a stretcher to a standing ovation, having been checked on by the anxious Johnstone and manager Nathan Jones.
Following the resumption of play, West Brom increased their lead further in the 66th minute. Robinson won the ball high and slipped Mowatt in. He in turn fed Grant, who received the ball with his back to goal; he turned and nonchalantly lifted the ball over Sluga.
The away side, fresh from seeing off newly promoted Peterborough at Kenilworth Road in their first league game of the season, were not interested in lying down so easily - though there was an element of fortune to their goal in the 74th minute.
Conor Townsend was guilty of taking too much time to clear his lines in his own penalty area, and his attempted clearance ricocheted straight across the face of goal and into the path of Cornick, who was not going to miss from close range.
Sixteen minutes were added onto the end of the second half, as a result of Clark's injury, That lifted the visitors, who enjoyed their best spell of the match in that period.
Another case of defensive communication between Kyle Bartley and Johnstone saw the ball bounce off the head of the former and, despite the best efforts of Dara O'Shea, Mpanzu converted from close range.