United went through to the fourth round of the Capital One Cup, knocking Liverpool out in the process, thanks to Javier Hernandez’s 46th-minute goal.
Luis Suarez, in his first game back following his 10-match ban for biting, gave his all to get the Reds back into the game, but they could not penetrate the United defence.
Manchester United will face Norwich City at home in the next round.
The Red Devils have suffered League defeats at Manchester City and Liverpool, but Moyes insists he will “turn it around”.
“It is not frustrating because I will sort it. I will fix it. I will turn it around. I have got no doubt about that,” he said.
“What have I learned? We know we have to improve. I am still getting to know the players. They are a really good team, they won the Premier League last year, but we need to make sure we improve.
“There was certainly some relief as it was a big day for us. We lost at Anfield a few weeks ago, we lost at the Etihad last week, so we wanted to put that right. I thought it was a really good game, an exciting game of football from both teams.
“In truth, I didn't think there was much in it. If you look at the two games, they have both been won by a set piece.”
Moyes made eight changes from the team that turned out on Sunday, including Nani, Hernandez and Shinji Kagawa in the starting XI.
“We made a lot of changes to make sure we made full use of the squad, to get them some playing time. Liverpool have started the season well. It was not a risk, with the players we have at Manchester United.
“They are all internationals and first-team players. There are no reserves in this squad. Everyone one of them could play in the first team. Some of the performances were excellent.
“What was important was that the whole club showed we were hurt by the game on Sunday. We did not perform to our levels. But they have gone out and the players have done a great job to get us a big result. I am really pleased with them.”