Rodgers was unhappy that Dowd awarded two second-half penalties, the first of which also saw Daniel Agger red-carded for a professional foul on Albion striker Shane Long.
He said: "I thought the scoreline was harsh. Congratulations to West Brom and Steve Clarke getting his first win as a manager.
"I've no complaints about the first goal, a fantastic strike from Gera, showing wonderful technique.
"But in the second half, although I'm not going to go on about the referee, I thought the two penalty decisions were very harsh.
"I don't think either was a penalty and Martin Skrtel doesn't even connect for the second one.
"With the first one, Long's pace has got him there. There is slight contact but I know Long well, how big and strong he is, and it was very harsh.
"With 10 men and then chasing the game, it becomes difficult."