Kevin Phillips' injury-time header snatched victory from the jaws of defeat for Tony Mowbray's side and propelled them into the top four of the Coca-Cola Championship.
Albion had trailed 2-1 with two minutes remaining when Phillips scored a controversial equaliser after Luton had wanted play stopped while David Bell lay on the ground injured.
Then the 33-year-old sent The Hawthorns wild with delight when he headed home Jason Koumas' free-kick to secure a sixth successive win for the home side.
Luton looked to have bounced back from the disappointment of losing top scorer Rowan Vine to Birmingham in a £3million deal 24 hours earlier.
Their first away win in the Championship since late September seemed secure when Warren Feeney put them in front with 20 minutes remaining. But that was before the intervention of the evergreen Phillips.
West Brom dominated the early exchanges in the swirling rain and midfielder Richard Chaplow came close to putting them ahead after three minutes when his rising 20-yard drive clipped the outside of the post after Phillips teed up the chance.
But Luton started to come more into the game and the pacey Feeney had a low drive deflected inches past the far post.
Feeney then should have done better when he side-footed wide from close range after being the first to react to a low cross from Dean Morgan.
Albion midfielder Koumas collected his fifth booking of the campaign when he chopped down Ahmet Brkovic and he will be suspended for next Saturday's clash at Leeds.
The home side suffered a setback when skipper Curtis Davies limped out of the action with a foot injury - he was taken to hospital for x-rays - to be replaced by Jared Hodgkiss.
Phillips was looking lively and tested Dean Kiely with a first-time effort from the edge of the box but Luton looked dangerous when they broke down the left flank.
Steve Robinson was within inches of connecting with a probing cross from Feeney after sliding in.
But in injury time Koumas received the ball on the left, powered in between two defenders and curled a right-footed drive into the corner of the net.
Albion dominated the early part of the second period and Phillips had the ball in the net only for his effort to be ruled out for a nudge by Diomansy Kamara on Kiely.
Phillips had a close-range shot blocked by Kiely after good play by Kamara who put the rebound over the bar. A flick-on by John Hartson set up another opportunity for Phillips but his attempted lob was off target.
Then, against the run of play, Keith Keane put Luton level after an hour. He exchanged passes with Robinson before drilling a left-footed drive past the dive of Russell Hoult into the corner of the net.
Hartson was marginally offside when he had a close-range effort disallowed before Luton stunned the home side by grabbing the lead after 70 minutes through Feeney.
Bell's incisive pass picked out the run of substitute Adam Boyd and his low cross was turned home by Feeney.
Bell could have wrapped the game up but his back-heel clipped the post.
It proved a costly miss as Phillips' double late strike turned the game on its head.
Teams
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).