Fabio Borini's 65th-minute penalty condemned United to a third consecutive defeat for the first time since May 2001 as the relegation-threatened Black Cats secured a 2-1 lead to take into the second leg of their semi-final shoot-out at Old Trafford on January 22.
A draw in a fortnight's time will be enough to send Gus Poyet's side into the final, but Moyes, who was without the injured Rio Ferdinand, Wayne Rooney and Robin van Persie at the Stadium of Light, is convinced better times are only just around the corner.
Asked how he was getting through an intensely difficult spell so early in his reign, the United boss said: "Well, because I have been here before.
"I have done it before, I'm experienced and I just sense that it's just on the verge of turning. I sense that we are about to get a bit better.
"If you are going to be in this job, then that's what happens. You have got to win and in the last few games, I have not won.
"We are going through a difficult time. It's part of football management, it's never going to be an easy ride."
Asked if the current situation represented the biggest challenge of his career to date, Moyes said: "It was the biggest challenge getting this job. To be the manager of Manchester United is a big challenge, whether it was on the back of wins or defeats."
Borini's spot-kick ultimately won the day for Sunderland after substitute Adam Johnson had been tripped by Tom Cleverley.
His contribution came 13 minutes after United skipper Nemanja Vidic had dragged the visitors back into the game with a bullet header which cancelled out Ryan Giggs' own goal in first-half stoppage time.
The Welshman could only turn former team-mate Wes Brown's cross into the net under pressure from another old boy Phil Bardsley on a night when United enjoyed plenty of possession, but could not make the most of it.
Moyes, however, was unhappy with referee Andre Marriner after he awarded the penalty on the advice of assistant Peter Bankes.
He said: "We had a player booked for that the other day. We had a player booked against Tottenham for the same.
"The referee is looking at it - he's in a great position to look at it, right down the line, probably 15 yards, not too close, but the linesman gives it from behind.
"He can't see, he can't see through Patrice Evra. But that's what we are finding at the moment.
"We defended a terrible free-kick (for the first goal), but how the referee has given a free-kick for that is way beyond me. I just cannot see how that is a free-kick."
Sunderland counterpart Poyet was thrilled with his side's display, which gives the club a fighting chance of reaching a first major final since 1992 and a first League Cup final for 29 years.
The Black Cats have now won six cup ties this season and only three Barclays Premier League games, and they have beaten Newcastle, Everton, Manchester City, Chelsea and now United while succumbing to less taxing opposition.
Poyet said: "That is something I need to find out, why we are capable somehow of beating top teams and to be so clinical in the cup, and we cannot do the same in the league.
"Analysis is part of my job, so I am going to keep looking at different things that we are doing in the cup. It could be the opposition, it could be how high you defend, it could be the mentality of the group of players...
"I need to find the reason because if not, something is happening, it's mental. But I need to find that reason."
But when he was asked who were favourites to reach the final, the Uruguayan replied: "No chance. Manchester United, with probably Rooney and Van Persie coming back in two weeks' time?
"Where we are is in a better situation because we won the first of the two. But playing Manchester United away, anything can happen, so we need to make sure we put in another very good performance."