After falling behind early on, United came roaring back on Sunday with goals from Marouane Fellaini, Ashley Young, Juan Mata and Chris Smalling.
The result ends a run of four straight derby defeats for United and moves them four points clear of City in the Premier League.
“The result is fantastic,” United manager Van Gaal told Sky Sports. “We have made big steps in the last six games because it was very difficult for my team to stand up and maintain our confidence in the philosophy after the (FA Cup) defeat to Arsenal.
“But I have to say, full credit to the players because they have to do it under pressure and the expectation of the fans. They were expecting we would win today and we can say we have, but it was not so easy.
“I am very happy for the fans because I have seen what it means to them. They are always supporting us even when results were not so good so I am very happy for them.”
The win is United’s sixth in a row in the league and Van Gaal was pleased with his side’s response after falling behind in the eighth minute.
“We started very badly, we were not the Manchester United we have seen in the last few matches,” he said. “We were very nervous and I don???t know why but we didn???t get pressure on the ball at the start.
“Then we scored the goal and from that moment I think there was more confidence in the team and we started to develop our pace and our gameplan. The second half was much better than the first. Until the moment Michael (Carrick) went off we were fantastic.”
Carrick was taken off late in the game with an injury, leaving United with 10 men as they had made all three of their substitutions.
“That was not so smart,” joked Van Gaal. “I wanted to give minutes to my players who are working very hard and some players didn???t know we were playing with 10 men as we couldn???t get the message on.
“It???s very difficult to get the message on with the noise.”