Albion, who remain fifth after their fifth successive draw, will be out of the top six if both Blackburn Rovers, who are home to Sheffield United at lunchtime, and Watford, who visit Sheffield Wednesday later, both win on Saturday.
The Baggies are next in action on Thursday night with a home game against Burnley, while Luton have another key home game the previous night on Wednesday against fellow strugglers Cardiff.
Luton Town boss Rob Edwards told BBC Three Counties Radio:
“A point was what we got and we have to take it but we deserved more. The momentum was with us the whole time. And we hit the woodwork with Hashy.
“To concede at the time we did just before the break made us make some changes and we had to take a few more risks, but that's what we're paid for.
“We’ve had three really good performances here at the Kenny in a row – and I really think we should have won all three.
“It’s difficult for all of us at the moment but the fans have stuck with us.”
West Bromwich Albion boss Carlos Corberan told BBC Radio WM:
"In general, I was happy with the defence. I was happy to achieve the point.
"You can criticise the way we tried to play. But you cannot criticise the effort.
"We just need to play with more quality."