Chelsea fans chanted throughout the match for Moyes to stay in his job, piling pressure on the Scot whose team have slipped 14 points behind leaders Arsenal as the manager attempts to fill the large shoes of his predecessor Alex Ferguson.
"It's a massive challenge. I'd hoped we would be in a far better position but we're not," Moyes told reporters.
"So the challenge is there for me. But I think that if you're a football person, it's a great one to have. I'm really looking forward to it."
Although United are 14 points adrift of leaders Arsenal after 22 matches, Moyes was not giving up hope of retaining the title.
"What we won't do is throw the towel in until we can't get there. The job is to try and finish first," he said, batting aside suggestions that United are in crisis lying seventh in the table.
"There's no clever answers except trying to win the next game," he said.
Moyes lamented his team's poor finishing after they wasted early chances and the careless defending that allowed Chelsea striker Samuel Eto'o to score a hat-trick.
Chelsea coach Jose Mourinho sympathised with his opposite number, saying United started as the stronger side.
"It's hard for them to be playing so well as they were in the first 20 or 25 minutes and be losing," the Portuguese said.
But Mourinho said he thought United were now too far behind the league's three front-runners to bridge the gap.
"I don't think David will be upset with me if I tell the reality, which is it's a 14 points difference (with Arsenal) and 13 and 12 (with Manchester City and Chelsea). Can they recover to one of these teams? They can," he said.
"But to recover to three of them, it needs three teams to have almost a collapse. I think for the title, it will be difficult for them."
(Editing by Ed Osmond)
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