Graham Potter's side looked more threatening early on. Neal Maupay converted Adam Lallana's cross only for his effort to be ruled out for offside, before Leandro Trossard’s shot was saved.
Five minutes before half-time Tariq Lamptey caused problems in the West Brom area and Branislav Ivanovic's attempted clearance struck Jake Livermore before rebounding into the net.
The Baggies pushed for an equaliser after the break, Matheus Pereira shooting wide and then striking a post, and on 83 minutes Grant lashed in a fine finish to level.
Brighton stay 16th with five points, two ahead of West Brom, who lie a place further back.
Brighton: 1 Nov v Spurs (A)
West Brom: 2 Nov v Fulham (A)
Budweiser King of the Match: Mat Ryan