The result left both sides close to the relegation zone, with Brighton 16th on five points from six games and West Brom 17th on three points as their winless start continued.
Livermore was unfortunate to get on the score sheet at the wrong end in the 40th minute as a desperate clearance by team mate Branislav Ivanovic cannoned off the midfielder's face into the back of the net after Tariq Lamptey's cross.
Visiting keeper Sam Johnstone had denied Leandro Trossard and Neil Maupay before West Brom fell behind, with the latter also guilty of poor finishing in the 30th minute when his first touch let him down.
West Brom pressed after the break and twice came close as home keeper Matthew Ryan saved a piledriver from Filip Krovinovic before Matheus Pereira shaved the outside of the post with another long-range effort.
Grant, who joined West Brom from Huddersfield Town during the close season, levelled with his first goal for the Baggies when he took a deflected pass into his stride and beat Ryan with an unstoppable shot into the top corner in the 83rd minute.