West Brom 17th on three points as their winless run began, and with Brighton 16th on five points from six games, the results left both sides close to the relegation zone.
After Tariq Lamptey’s cross, a desperation clearance by teammate Branislav Ivanovic cannoned from the midfielder’s face into the back of the net Livermore was unlucky to get on the score sheet at the wrong end in the 40th minute.
Before West Brom fell behind, visiting keeper Sam Johnstone had thwarted Leandro Trossard and Neil Maupay, with the latter still guilty of bad finishing in the 30th minute when he was let down by his first touch.
As home keeper Matthew Ryan saved a piledriver from Filip Krovinovic before another long-range shot allowed Matheus Pereira save the outside of the post West Brom pushed and came close twice after the break.
During the close season, Grant, who joined West Brom from Huddersfield Town, levelled with his first goal for the Baggies when he took a deflected pass into his stride and beat Ryan in the 83rd minute with an irresistible shot into the top corner.
West Brom – Brighton 1-1
Goals: Livermore(40), Grant(83)