Early goals in each half from Alex Oxlade-Chamberlain and Lukas Podolski put Arsenal in control before Steven Gerrard pulled a goal back from the spot a minute before the hour mark.
However, Rodgers was angered when referee Howard Webb failed to award a second penalty after Luis Suarez was brought down by a clumsy Oxlade-Chamberlain challenge.
"I thought the best team lost, we probably should have been two up within six or seven minutes," he told BT Sport.
"Going forward we were outstanding but we're disappointed with the goals.
"We're bitterly disappointed not to get something (from the game) and not to get a second penalty.
"It was probably more clear and blatant than the first one. I'm not sure whether it's because it was just after (the first penalty).
"Luis has got a touch on the ball and Alex has just taken him out inside the box under the referee's eyes."
It leaves Liverpool with just the Premier League to focus on while Arsenal remain in contention for titles on three fronts - with their UEFA Champions League campaign continuing against Bayern Munich on Wednesday.
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