Rodgers said: "We are disappointed with the result, but sometimes when you are beaten by a better team, you have to accept that.
"It was a game they deserved to win. They had greater control, but we were always in the game because of our ability to create chances."
The outcome, however, could have been different but for a disallowed Liverpool goal in the first half which would have made it 1-1.
After being pushed over by Bacary Sagna, Suarez took a quick free-kick which released Daniel Sturridge who crossed for Jordan Henderson to tap into the net - only for referee Martin Atkinson to pull play back and book the Arsenal defender.
Rodgers was not impressed. The Northern Irishman said: "I got an explanation from the fourth official on it, and it is disappointing, to be honest.
"He said it was the next phase of the game, so he was given a yellow card, and it could only have been given at that point.
"I thought it was a poor decision because you have to let the game flow, but that was the reason he gave it. It was clever play (from us), and we got a goal from it."
Rodgers added: "There is a long way to go. We have made an excellent start and I can't ask any more from the players. They are pushing every game."
Defender Glen Johnson was missing through illness, which meant Jon Flanagan came into the side.
Rodgers said: "Half an hour before I named the team, Glen had to pull out.
"He had to go to hospital with an illness. We hope he will be out (of hospital) this evening."