The Foxes took the lead, but Arsenal overran the hosts to win 3-1.
Rodgers said, “I thought we didn't play well in the first half. We didn't have the ball enough. I expected us to pressure them more. We were too static; I thought we looked a little tired to be honest.
“In the second half we showed more spirit but we were unlucky with the third goal. Credit to our players, they kept working and gave everything. We just lost to the better team today.
“Arsenal are able to rotate their squad. We tried to rotate our team on Thursday. It's not ideal [having matches so close together] but we have to play the games. We just get on with it.
On injuries to his players, Rodgers added: “I think Jamie Vardy is okay. He felt his calf go making a pass but it was fine for the rest of the game.
“It looks like Harvey Barnes is going to need an operation on his knee. I just spoke to the doctor; he might be out for six weeks. That'll be a huge blow for us.
“It's been unfortunate but it's the uniqueness of the season. We haven't cried about it all season, we're not going to start now.
“We have to find a way with the players that we have. The players have been on a brilliant run over the course of the season. Twelve games to go, those influential players are out. The challenge is to find a way to keep [our form] going."