Life after Sir Alex Ferguson was always going to be difficult, but few could have predicted United being eight points off the pace after nine matches.
They remain eighth in the Premier League after Saturday's 3-1 defeat of Fulham, although Moyes' side are slowly quietening the naysayers.
The victory at Craven Cottage was their fourth in succession in all competitions and the first time United have secured back-to-back league wins under the former Everton manager.
"We knew we would have to pick up points as we went along and I was always confident that would happen," Moyes said.
"It was a good victory, an important victory for us. It gets you three points and hopefully we will try and continue doing that and keep the momentum building.
"Our job is to climb the league table. We want to get the nearer the top and I have got no doubt that by the end of the season we will.
"Where we are right now I don't think is really that important, except that you don't want to let the teams get too far away from you and we've got to try and make sure they don't.
"We're going to have keep building on today's performance and the performances of the last week or so."
United's win came courtesy of three goals in 13 first-half minutes, with dodgy defending and clinical finishing allowing Antonio Valencia, Robin van Persie and Wayne Rooney to net.
Fulham pulled one back when Alex Kacaniklic's shot deflected in off Rooney, with the hosts impressing in a second half in which United started with the enforced substitutions of Tom Cleverley, Jonny Evans and Rafael.
"Tom Cleverley had double vision, Jonny Evans' back stiffened up and Rafa went over on his ankle on the touchline," Moyes said of the triple change.
"It forced me into three changes at half-time, which disrupted the momentum of the team a little bit and the fluency of the team. I was pleased with the lads that came on."
Moyes is hopeful the trio will be available for Tuesday's Champions League clash against Real Socieded, although concedes Rafael is the biggest doubt.
The injuries were the only negatives in a fine afternoon for United, whose strikers Van Persie and Rooney impressed once again.
"I thought there was great combination play between them," Moyes said. "I think they both made the goal for each other.
"There was good link-up play, so I am just starting to see really good signs from the two of them.
"It was great that they both got on the scoresheet because that is what forwards do and I have always said we need these boys to get a hatful of goals if you're going to be successful.
"They are both incredibly talented and both understand each other and the talents one another have got, so they are both excellent players."
Moyes also praised Fulham for the way they came out in the second half, coming close on several occasions to adding to Kacaniklic's strike.
The Scot's kind words, though, were nothing compared to the anger aimed at Jol by Fulham fans.
"Jol out" was chanted on several occasions against United, making his position at the helm look increasingly precarious.
Asked how he felt on the touchline after Moyes' side scored their three goals, the Dutchman said: "Lonely, very lonely.
"We had aspirations, we were ambitious before the game. We really wanted to do well.
"I've got a great team with a great team spirit. Dimitar Berbatov was skippering us again, so he wanted to win as a captain.
"The players wanted to win for him and maybe a little bit for me and for the club. To concede goals like that looked horrendous."
As if a third straight loss was not bad enough, Fulham may well now lose Sascha Riether to a ban after a late stamp on Adnan Januzaj was not punished by referee Lee Probert - an incident missed by Moyes and Jol.
"That would be disappointing," Jol said about a potential ban. "If it is true, then we have to talk to him.
"I have got some good back-up players as well, so that is not a disaster. The result is a disaster."