Liverpool made a bright start at Emirates Stadium but Santi Cazorla fired the Gunners ahead in the 19th minute and it was the home side who created the better chances for the remainder of the game.
Aaron Ramsey sealed victory for Arsenal with a magnificent second and Rodgers had few complaints about the result after seeing his side suffer only their second defeat of the season.
"I thought the best team won. Their mobility and passing was better than ours and we could've done better with the two goals," Rodgers told Sky Sports, but he nevertheless praised his side for their fighting spirit.
"We kept going all night, the players have been brilliant in their application. They went right to the end.
"We created chances, we just couldn't put the ball in the net."
Despite the loss, Liverpool can still reflect back on a fine start to the season and Rodgers said: "I said that after 10 games it's a good analysis, you can look at the league and see where you're at.
"I think what we can see is my players have been brilliant in this opening period, we're gonna be in with a shout.
"We understand the competition is great, but so far we've shown enough that we can be in with a fighting chance of the top four and that's what we'll be aiming to do.
"We're in that top four, we've just got to put this game to bed, learn from it and win our next game."