In a match that the Reds dominated, Gareth McAuley and Romelu Lukaku netted late on for the Baggies after Steven Gerrard missed a penalty but the Northern Irishman praised his players' efforts after the final whistle.
"The players gave everything," he told Sky Sports after the game. "It was one of those games where we just couldn't make the breakthrough.
|20/1||Liverpool are 20/1 with BetVictor to beat Zenit St Petersburg 2-0 in the Europa League
"We weren't at our best but we were certainly on the front foot and looking to create. Full credit to West Brom, [Ben] Foster's made some brilliant saves. It was a good away performance from them and the longer it was 0-0, the riskier it got.
"I can't blame Gerrard for missing the penalty; he had a fantastic game and he had the courage to step up and take it."
The Reds had won their previous three games at Anfield, while drawing at Manchester City and Arsenal in their last two fixtures, and Rodgers is keen to get his side back into form, lamenting: "I'm just disappointed that our good run, particularly at home, has come to an end.
"We gave everything [on Monday] and the keeper had a fantastic match but we'll go again; we've got an important game in the Europa League on Thursday and we'll be ready for that."