Kevin Phillips scored two late goals to snatch victory for West Brom.
Jason Koumas cut in from the right and beat two players before brilliantly opening the scoring for WBA before Keith Keane equalised from 18 yards.
Warren Feeney then put Luton in front, slotting home Adam Boyd's low cross from close range.
Phillips resisted several attempted tackles to equalise from 10 yards late on before dramatically heading home a free-kick in injury time.
"That said, we have been averaging three goals a game at home recently.
"Luton made life difficult for us and we have got to give them a lot of credit. It was an open game."
606: DEBATE
"We have gone soft because of all the foreigners that have come over here.
"As soon as they are touched they go to ground. I will never have any of my players feigning injury.
"We will continue to play the football that we are capable of and hopefully results will start to turn for us."
West Brom: Hoult, Watson, Clement, Curtis Davies (Hodgkiss 19), Robinson, Kamara (Koren 74), Chaplow, Greening, Koumas, Hartson (Gera 74), Phillips.
Subs Not Used: Zuberbuhler, Carter.
Booked: Koumas.
Goals: Koumas 45, Phillips 88, 90.
Luton: Kiely, Keane, Coyne, Perrett, Davis, Brkovic (Boyd 66), Bell, Robinson, Emanuel (Langley 66), Morgan, Feeney.
Subs Not Used: Beresford, O'Leary, Stevens.
Booked: Langley, Bell.
Goals: Keane 60, Feeney 70.
Att: 19,927
Ref: J Moss (West Yorkshire).