The defeat left Palace winless in four games and 14th in the table. They are now on their worst run since Patrick Vieira was sacked in March.
Ahead of his side hosting Liverpool, Hodgson, 76, said: “I’m a bit distressed by it, certainly distressed by my choice of words, which I bitterly regret. ‘We’ve spoilt the fans in recent years' — I don’t think we have. I think they’ve been very patient actually.
“They’ve been so good to me and the team during the almost-five years I’ve been at the club now. I would be devastated if they seriously thought I don’t appreciate the way they do try to get behind the team at all times.
“We are certainly going to need the fans tomorrow. The fact that we are at home is one of our very big hopes. When things aren’t going well, that’s when you really do appreciate a fanbase like we have.
“I felt so exasperated after the game. Basically, I felt like booing myself at one stage, so that’s the way things are. I owe them an apology.”
Speaking about the Liverpool game, he said: "I am fearful that we won’t be able to win it — but I still believe we can."
Palace face a tough run of fixtures against Liverpool, Manchester City, Brighton and Chelsea over the Christmas period, and while Hodgson is not thought to be at immediate risk of losing his job, results must improve.
“We’re not panicking ourselves, the players and the coaching staff,” he said when asked whether the club’s board might be panicking about their poor form.
“They are questions you have to ask the people higher up, who might want to panic — I don’t know. But I don’t get the feeling that that’s the case. Certainly, the chairman here is a very experienced man. He has seen this before. If there is panic, it’s not being filtered down towards me and the team.”
Hodgson confirmed left-back Tyrick Mitchell would not be fit to face Liverpool after being substituted with a muscle strain in his hip against Bournemouth.
“He’s injured too,” said Hodgson. “This game was definitely too quick coming around for him to be considered, but that doesn’t mean to say he won’t be considered for the next game.”