United are 14 points behind Barclays Premier League leaders Arsenal after a 3-1 defeat to ]Mourinho's title hopefuls.
Phil Jones backed off Samuel Eto'o, allowing the Cameroon international to score a deflected opener, and the striker took advantage of shoddy marking from two set-pieces to complete his hat-trick on what was a sorry afternoon for the Red Devils.
United never looked like mounting a comeback from 3-0 down like they did here two years ago and Javier Hernandez's 78th-minute goal was a consolation they barely deserved.
Mourinho said after the game that United's rivals would have to suffer dramatic collapses for Moyes' team to reclaim the title, but the Scot disagrees.
"We won't throw the towel in until we can't get there," the United manager said.
"The job is to try and finish first. I will keep trying to do that.
"Before today we'd lost one in six in the league, and I didn't think there was a big difference between the teams today, not at all.
"The difference was our defending to set-pieces. Nothing else."
Moyes always knew succeeding Sir Alex Ferguson, who enjoyed 26 years of success at Old Trafford, would not be easy. But he conceded the performances of his team have not been up to scratch so far.
United looked ordinary at times on Sunday. Mourinho's men were more compact, they looked far more incisive on the attack and unlike United, their defensive stability never seemed in doubt.
The table does not look pretty for Moyes. United sit in seventh, just one point ahead of Newcastle. They are six points adrift of the top four and that margin will increase by one more point if Everton beat West Brom on Monday.
One reporter put it to Moyes that his team were in crisis.
Again, the Scot disagreed: "That's your opinion. Crisis is your word, not my word."
Moyes admitted he knew he was embarking on a "massive challenge" when he took the giant step up from Everton to United last summer.
Some United fans wanted the man who was in the home dugout to succeed Ferguson due to his experience on the biggest stage.
But Moyes still feels he is the man to turn things around.
When asked how his confidence was holding up following the latest defeat to a rival, Moyes said: "Fine. It's a difficult task, but (it's about) perseverance and keeping doing what's right.
"I thought we did a lot of good things today. We have players to come back, and this is a project I know that I'm going to improve as it goes along.
"I was hoping to win more and be competing a bit more than we've been, but that'll come."
Just as was the case in the 4-1 Manchester derby defeat in September, United's marking - particularly at set-pieces - was poor.
A weak clearance allowed Ramires to work the ball to Gary Cahill and he picked out the unmarked Eto'o for Chelsea's second.
Cahill was allowed a free header in the run up to the third goal, which Eto'o poked home after escaping the clutches of Antonio Valencia, who man handled the striker on the line before allowing him to spin and finish.
"The game came down to two set-pieces and we didn't defend them well today. That was probably the difference," Moyes added.
"While we were 1-0 down we still had a chance. We were still in it but we gave away a free kick that led to a corner kick and we were terrible at defending the corner kick.
"We headed it out, we got rid of it and we didn't defend it well enough in the second phase and we found ourselves 2-0 down at half-time."
Moyes' afternoon went from bad to worse towards the end of the match when Nemanja Vidic was sent off for a bad challenge on Eden Hazard.
Rafael was guilty of an even worse two-footed challenge on Cahill moments later, although the full-back connected with the ball rather than the player.
Luckily for the Brazilian, and Moyes, referee Phil Dowd gave the right-back only a yellow card.
"I thought Vidic wasn't a sending-off, but I've seen Rafa's and I think that could have been a sending off," Moyes said.
The only crumb of comfort for Moyes was that Robin van Persie and Wayne Rooney are set to return to training next week.
"I hope so," Moyes said when asked if they would return to the fold on Monday.