The Anfield boss watched his side lose at the Emirates but he remains confident that they have what it takes to qualify for next year's Champions League
The Reds missed the chance to overtake the Gunners at the top of the Premier League on Saturday, as goals from midfielders Santi Cazorla and Aaron Ramsey inspired Arsene Wenger's side to victory.
The Anfield outfit had been mentioned as title contenders following a strong start to the season, but Rodgers' ambitions are more conservative.
"I think what we can see is that my players have been brilliant in this opening period," he told Sky Sports.
"I think we're going to be in with a shout. We'll be in with a fighting chance of that top four and that's what we're aiming to do."
Luis Suarez and Philippe Coutinho both had chances to mount a comeback at the Emirates Stadium, but Rodgers did not believe his side deserved anything from the game.
He added: "I thought that the best team won; they had a greater control of the game. We still had chances.
"Their mobility and passing was better than ours; we could have done better with the two goals. We're disappointed: we never pressed the ball well enough; we were never tight enough.
"We kept going all night; the players have been brilliant in their application of the game. We created chances but just couldn't put the ball in the net."