The Reds made two changes from their 7-0 win over Crystal Palace, as Curtis Jones returned in place of the injured Naby Keita while Mohamed Salah was also recalled in place of Takumi Minamino.
Liverpool did not give West Brom time and space to breathe in the opening minutes and Sadio Mane made their pressure count in the 12th minute after controlling ball from Joel Matip before rifling the ball beyond Sam Johnstone.
Sam Allardyce's charges offered better resistance after going goal down but they were not growing enough in the game to create chances.
Salah had an opportunity to double the lead for Jurgen Klopp's troops eight minutes before the break with a half-volley from the edge of the box that missed slightly over the crossbar.
West Brom continued to keep Liverpool at bay after the break and the hosts had defensive worries on the hour-mark after Matip went down injured and was replaced by youngster Rhys Williams, who partnered makeshift centre-back Fabinho.
The visitors had a great chance to level matters in the 73rd minute after Karlan Grant raced past Williams to go clear on goal from a ball over the defence but his low shot was saved by Alisson.
West Brom eventually found the equaliser in the last eight minutes after Semi Ajayi rose the highest in the box to power a header in off upright, leaving Alisson with no chance.
Roberto Firmino came very close to snatching the winner for Liverpool in the last minute with a free header at the back post that was kept out of goal by the fingertips of Johnstone, as Allardyce's men held on for a point.