Rowan Vine's fifth and sixth goals of the season had seen Hatters take the lead in each half, only to be pegged back by strikes from Darren Carter and substitute Zoltan Gera.
Town saw a host of chances go begging in a one-sided first half, although the Baggies did hit the woodwork through Jason Koumas just before the break and had a goal disallowed in injury time at the end of the game when Ronnie Wallwork's effort was chalked off for a foul.
Town started at their usual frenetic pace and forced the visitors' Swiss international stopper Pascal Zuberbuhler into action within the first minute when Chris Perry deflected Richard Langley's right-wing cross into his goalkeeper's arms.
It was Luton who dictated the pace as Dean Morgan blasted over left-footed from 12 yards in the ninth minute after Vine's left-wing cross had deflected into his path.
Full-back Kevin Foley was next to go close in the 20th minute when his 30-yard run and resulting 20-yard shot flew inches past the post off Paul McShane, before Luton finally and deservedly took the lead six minutes later.
Richard Langley appeared to have lost possession at first in midfield, but won the ball back and played in Vine who accelerated past McShane and buried a left-foot shot straight into the bottom corner from the edge of the penalty area.
Vine almost got his second of the afternoon just after the half-hour when his instant turn and 20-yard shot from Edwards' pass was pushed around the post by Zuberbuhler.
Town looked rampant, but instead the visitors equalised with a goal out of nothing in the 34th minute when, after Sam Parkin had lost the ball on the halfway line, Nathan Ellington found Carter, who rifled home an unstoppable effort into the corner of the net from 22 yards.
West Brom were level for less than a minute however as, straight from the restart, substitute Gera fouled Parkin on Luton's left and from Morgan's low, whipped in delivery, Vine touched home from six yards.
The Baggies almost levelled out of the blue again in the 39th minute when Koumas managed to bring down a bouncing ball on the edge of the area before smashing a 20-yard effort that ricocheted away off the post.
West Brom livened up after the break, but it didn't take Luton long to get back to their attacking first-half style as man-of-the-moment Vine robbed Wallwork in the 49th minute, before smacking a rising 25-yard drive off the top of the crossbar.
The Baggies though came back again and Marlon Beresford firstly stopped Ellington's fierce 20-yard strike at the second attempt in the 52nd minute, before producing a breathtaking reaction save a minute later to tip Kevin Phillips' close-range prod around the post.
Edwards was upended by Robinson after a mazy dribble into the box in the 59th minute, but Town's penalty appeals were ignored by referee Phil Dowd.
The howls of protest from home Luton fans turned to moans of disbelief as West Brom levelled in the 61st minute and to rub salt into the wounds it was Robinson who made the goal. The full-back's left-wing cross was allowed to travel across the face of goal for Gera to net at the far post.
West Brom thought they had won the game in injury time when Wallwork swept the ball home following a Koumas corner, but the goal was disallowed for a foul by McShane on Luton sub Warren Feeney.