Rodgers was unhappy that Dowd awarded two second-half penalties, the first of which also saw Daniel Agger red-carded. Although Long's resulting penalty was saved by Jose Reina, Peter Odemwingie tucked away the second spot-kick, awarded for Martin Skrtel's foul on Long, while Albion also found the target through Zoltan Gera and Romelu Lukaku.
Rodgers, however, felt the margin of victory did not reflect his side's efforts. He said: "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."
He added: "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."
The former Swansea boss admits his Liverpool side remain a work in progress but still saw some encouraging signs in the opening 45 minutes.
He said: "We showed some signs of our quality but the penalties and the sending-off kills you, especially a new team coming together. But I've got to give big credit to the players. They ran to a standstill today and I can't complain.
"It is still a work in progress. It is very much a marathon we are in. This is the first game."
Meanwhile, Baggies boss Clarke insisted he gained no extra satisfaction from defeating Liverpool.
Clarke, who was number two to Kenny Dalglish at Anfield last season, said: "The manner of departure from Liverpool is not an issue. That's what happens in football these days. It doesn't give me any extra satisfaction winning this game. If I was sitting here and we had played someone else and won, I would be just as happy."
Clarke was unsurprised by the quality of his side's performance. He said: "I think I've been in the game long enough to know you enjoy these moments but you know there are different Saturdays to come. But the performance today showed these players are comfortable in the Premier League."
Source: PA