Liverpool had performed brightly in the first half, as Mo Salah opened the scoring with a close-range finish before Sadio Mane opened his account for the season just before the break.
The second half started with Liverpool having less dominance of the ball, but West Ham equally not looking like psing a serious threat to Alisson's goal.
The 'Reds' picked up the ball through Roberto Firmino just outside the opposition box, who ran at the defence with intent.
Mane was stood in a clearly offside position, but neverthless the assistant referee's flag failed to go up, allowing the Senegalese forward to turn quickly and drill the ball low in to the left corner of the net.