The visitors completed a League double over The Reds courtesy of goals from Gareth McAuley and Romelu Lukaku late on after Steven Gerrard had missed a penalty.
Without Daniel Sturridge, missing through injury, Rodgers opted to start Jonjo Shelvey in an advanced role and Luis Suarez up front. Liverpool lacked a presence in the box and around the final third and introduced Fabio Borini for Shelvey after an hour.
The defeat sees Liverpool move down to ninth, with West Brom moving above them in the League. They travel to St. Petersburg in the Europa League on Thursday before hosting Swansea at Anfield on Sunday.
The defeat also means Rodgers side has still not won a game against a team currently in the top half of the table.