An unfortunate own goal from West Brom's Jake Livermore gave Graham Potter's side a first-half lead but the Seagulls were unable to hold out for the win.
The home 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 their pressure paid off when an attempted clearance from Branislav Ivanovic struck Jake Livermore and rebounded into the back of the net.
Slaven Bilic's West Brom started the second half positively and had a Conor Gallagher effort blocked just after the restart.
The Baggies continued to put pressure on the home defence as the match wore on, forcing Mat Ryan into another save from Filip Krovinovic.
Matheus Pereira had a good chance for the visitor midway through the second half but he put his effort wide before later clipping the outside of a post.
But West Brom's second-half pressure finally paid off when Grant turned a Callum Robinson ball past Ryan in the 83rd minute to ensure that the Baggies would leave the south coast with a point.
The result, which leaves both teams in the bottom five, means Brighton has won just one of its past 12 home league games.
Brighton boss Potter said he was frustrated by the draw.
"We have to analyse why we were not so good in the second half, why our performance dipped a bit but now we have to give credit for the opposition for raising their performance," he told the BBC.
"We have had some good performances this season. There is more to do, we have to improve but something to learn from and build from."