Despite the score, Albion have been on terms with the Merseysiders.
A clear chance for Solly March very early on drew a save from Allison.
But that was upstaged by a great strike from Jordan Henderson to put Liverpool in the lead.
Sadio Mane headed in a second just before the half hour.
Yves Bissouma almost got Albion back in it with a super strike which hit the post.
Mane had a second ruled out by VAR for handball.
And then an apparent wonder strike from a tight angle from Enock Mwepu put Albion right back in it at the break.
Can Albion come back and win again at Anfield?
