United have little time to dwell on the disappointment of a seventh league defeat of the season with Sunderland heading to Old Trafford on Wednesday with a 2-1 advantage from the first leg in the lightly regarded competition.
"We're going to go out and do everything we can to get through," former Everton manager Moyes told MUTV on Sunday.
"I want to give the supporters something to cheer about. There's not been an awful lot and I'm fully aware of that but I thought how we played today was not bad.
"The players played well but if you are going to defend the way we did then we are going to find it tough."
Moyes will be forced into at least one defensive change with captain Nemanja Vidic serving a suspension after being shown a straight red card for a late lunge on Chelsea winger Eden Hazard, a tackle that epitomised United's frustration.
Another member of United's frail defensive four, fullback Rafael, then angered Chelsea with a dangerous two-footed tackle in the final stages that took the ball from Gary Cahill but resulted in a yellow card.
"I thought Vidic wasn't a sending off but I've seen Rafa's and I think that could have been one," Moyes told reporters.
"There were two tackles. People want their players to tackle. I thought one was a yellow card. On another day, the other was a red."
While Mourinho was dismissive of United's chances of overhauling a 14-point gap on Arsenal with 16 games remaining, Moyes refused to rule out United adding to their 20 English titles in May.
"What we won't do is throw the towel in until we can't get there," said Moyes.
"The job is to try to finish first. There are no clever answers. We'd lost one in six before today in the league and I didn't think there was a big difference between the teams today. The difference was our defending to set-pieces.
"Confidence is fine. It's a difficult task but it's perseverance.
"We have players to come back and this is a project that I'm going to improve as we go along. It's a massive challenge. I'd hoped I'd be in a far better position than this but we're not."
Forward pair Wayne Rooney and Robin Van Persie are likely to remain out of action for a few more weeks but Moyes' sole expensive recruit Marouane Fellaini is due back soon and will have a cast removed from his wrist on Monday.
While Moyes, who replaced Alex Ferguson in the close season, was supported by England midfielder Michael Carrick in refusing to rule out their slim title chances, Vidic did not believe United could overhaul the leaders.
"Yes, I think so now we are too far behind," the Serbian told Sky Sports.
"We have to focus now to get in the top three, top four. We have to be positive. It's not going to be easy but we have to now fight for different things.
"We have to focus on the (Capital One) Cup and the Champions League.
"It's never easy but we have to stay positive, we have to stick together, we have to work hard to change the things around.
"Hopefully we can play better football and with a bit of luck I think we can achieve that."
(Writing by Patrick Johnston; Editing by Peter Rutherford)
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