The Eagles dominated the encounter at Selhurst Park but had to wait until midway through the second half before Albion's resistance gave way.
Although the ball was cleared from beyond the line, referee Jonathan Moss made use of video technology to confirm that the goal indeed stood.
Nine minutes from time, Pape Souare came on for his first appearance since a car crash in 2016 in which he broke his leg and jaw and put his career under threat.
The final whistle saw Palace finish 11th in the table, a remarkable turnaround considering that the Eagles were bottom, pointless and goalless after seven games.
West Brom, meanwhile, finish bottom of the standings with 31 points from 38 games and will spend the summer preparing for life in the second tier.
CRYSTAL PALACE (4-4-2): Hennessey; Wan Bissaka, Tomkins, Sakho, Van Aanholt; McArthur (Benteke 68'), Cabaye, Milivojevic, Loftus-Cheek; Townsend, Zaha
WEST BROMWICH ALBION (4-4-1-1): Foster; Nyom, Dawson, Hegazi, Gibbs; Rodriguez (Sturridge 73'), Livermore, Brunt, McClean; Krychowiak (Chadli 61'), Rondon (Robson-Kanu 72')