The Baggies scored three goals in just over half an hour of play to make Brendan Rodgers' first match day in charge of the Reds a miserable one.
Returning from injury, Zoltan Gera opened the scoring with a brilliant strike from 20 yards late in the first half, taking a touch from a deflected corner, and then hammering he ball into a hapless Pepe Reina's top corner.
The visitors were then reduced to ten men just prior to the hour-mark when Daniel Agger pulled down Shane Long in the area as the last defender but the Irish striker was unable to convert from the penalty spot with Reina easily keeping out his weak effort.
However, Long's miss was soon atoned for by Peter Odemwingie in the 64th minute when the Nigerian beat Reina from the spot after referee Phil Dowd adjudged Martin Skrtel to have tripped Long in the box.
The Hawthorns, West Bromwich - 26,039
Red card:LIV - Daniel Agger 58'
Match notes: - Steve Clarke (first match in charge of WBA), Brendan Rodgers (first match in charge of LIV), Claudio Yacob (WBA & Premier League debut), Joe Allen, Fabio Boriin (LIV debut), Romelu Lukaku (WBA debut)