Grabban, who had only been on the pitch for five minutes, pounced on a terrible mix-up between Albion left-back Paul Robinson and goalkeeper Dean Kiely to steer the ball into an empty net four minutes into stoppage time.
It was a cruel blow for Albion who had fought their way back on to level terms despite playing for the final 50 minutes with ten men.
Neil Clement had given them the perfect start with his 26th-minute opener only for Clinton Morrison, who also hit the bar at the start of the second half, and a Ben Watson penalty to fire Palace into a half-time lead.
Martin Albrechtsen, who had been on the pitch for just two minutes after replacing Curtis Davies, was dismissed for conceding the penalty.
Despite their lack of manpower, Albion staged a tremendous second-half rally and were rewarded in the 76th minute when Kevin Phillips raced on to a pass from Diomansy Kamara before drilling home his 15th goal of the season.
But Phillips' heroics were undone by Grabban as Albion's promotion push veered further off course with a third straight defeat.
In contrast, Palace are making a late push to get into the play-offs. They have now won four of their last five games and collected 13 points from a possible 15.
Albion dominated the opening stages when Jason Koumas, Kamara and Jonathan Greening all went close before they got their reward through Clement's first goal since November 2005.
He drilled a low 20-yard free-kick through the Palace wall after Kamara had been fouled by Mark Hudson.
But all their good work was undone by a mad five-minute spell when Palace turned the game on its head.
Jamie Scowcroft picked out Morrison in the 35th minute who held off three challenges to fire Palace level from 12 yards with his tenth goal of the season - four of them coming in the last four games.
Albrechtsen, who had replaced skipper Davies when he was injured in the build up to Morrison's goal, then suffered his personal nightmare after pulling down Michael Hughes who had raced on to a Scowcroft header.
From the resulting penalty, Watson sent Kiely the wrong way from the spot.