Without Wilfried Zaha - who is isolating after testing positive for coronavirus - Palace laboured to create chances against a dogged Burnley side, but grew closer late on and were denied late on as Nick Pope threw himself in front of Christian Benteke's effort and needed treatment for doing so.
"I thought we played well second half," Hodgson said. "An unfortunate first goal, a misplaced header but apart from that we didn't allow them many chances to score a goal. In the second half they stood firm and held out for their victory.
"An error like that is always a possibility, you have to accept it and then you've got the rest of the game to put it right but we started a bit too late.
"That's all teams [lacking in the final third]. That's even teams of Manchester City's quality, let alone Crystal Palace's. I can't ask more in terms of effort and determination I am just disappointed we started a bit late. The longer the game went on the closer we got to that equaliser."
It wasn't just Zaha which Palace were missing. James McCarthy and Luka Milivojevic were also absent and Hodgson added that there is no guarantee their star man would have made a different.
"You always miss your quality players but there is no guarantee he would have had a great game tonight, or taken the chances or opened up Burnley any more than other players did," Hodgson said. "It is a long season, we will get players back and let's hope we can put this defeat behind us and get winning again."