The visitors were left stunned by an eight-minute spell just before the break in which the hosts scored three times at the Emirates Stadium.
It leaves Liverpool's UEFA Champions League hopes hanging by a thread and manager Rodgers offered a pragmatic assessment of their chances.
"It'll be very, very difficult now," he told BT Sport. "It was a game we needed to take something from and obviously it's a disappointing scoreline for us.
"Right from kick-off we were too negative. We put ourselves under a little bit of pressure and in first 10 minutes they had decent control of the game.
"I thought after that we were the better team and created the better chances. We were looking to go in half-time maybe 0-0 or in front, but then we defend very poorly for all three goals really."
Emre Can's late dismissal for a second booking compounded Liverpool's woes, but Rodgers did have some words of praise for Raheem Sterling after a week in which his Anfield future was cast into further doubt.
He added: "I thought Raheem was outstanding in all fairness. I thought the kid done great."
Liverpool's attention will now turn to Wednesday's FA Cup quarter-final replay at Blackburn Rovers.
