The Gunners were two goals up early in the second half after Alex Oxlade-Chamberlain and Lukas Podolski had found the back of the net before Reds skipper Steven Gerrard converted from the spot after Podolski had felled Luis Suarez in the area.
Liverpool had shouts for another spot-kick when Oxlade-Chamberlain appeared to bring down Suarez but referee Howard Webb waved away the protests, much to the dismay of Rodgers.
"We're bitterly disappointed not to get another penalty," he said. "It was probably more clear and blatant than the first one.
"The second one is clear. I'm not sure if it was just after [the first one being awarded], but it was a clear penalty. Luis got a touch on the ball and young Alex has taken him out.
"It was a strange decision. It was a penalty... and it was a clear one."
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