United had not scored a first-half goal in 11 games at Old Trafford and were booed off after losing 1-0 to Southampton in their previous home match.
But Jesse Lingard's 14th-minute header eased the tension on Tuesday and United went on to score a superb second through Anthony Martial before the break and a third courtesy of captain Wayne Rooney in the second half.
Playing down suggestions that it had been United's best display of the season, Van Gaal said: "No, I don't think so. The difference is the first half we score and then it is much more easy.
"We have played games for 90 minutes with a lot of pressure and also against an opponent who had a higher level than Stoke City tonight (Tuesday).
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"I think the players have made a great performance, not only defensively, but now also 'attackingly' because we created a lot of chances.
"We had fantastic finishing of team goals, building up from the back, which is what I like to see."
Although he did not get on the scoresheet, Juan Mata was a pivotal performer for United in a central position behind Rooney, creating several chances and having opportunities to score himself.
"He was participating in all the three goals so that is also what I like to see," said the United manager.
- Guardiola welcome -
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Rooney was dropped for United's 2-0 Boxing Day defeat at Stoke as United lost for the fourth game running -- a run that prompted serious scrutiny of Van Gaal's position -- but has since scored seven goals in as many games.
"I hope for him and it is also good for me then because when he scores, we win," said Van Gaal.
United's win was a welcome tonic following the announcement on Monday of Pep Guardiola's appointment as Manchester City's next manager.
City hope the arrival of the Bayern Munich coach will drive them to further success domestically and in Europe, placing further pressure on United, who are still five points away from a Champions League spot in fifth place in the Premier League.
He said: "He is my former captain so I am very pleased to see him and I can speak Spanish, so it is convenient for him."
Defeat for Stoke completed a disheartening sequence of results that had previously seen them dumped out of the League Cup and FA Cup in successive matches.
"The worst thing that can happen at Old Trafford is to concede early," said Stoke manager Mark Hughes, the former United striker.
"That is the same team that did exceptionally well against Liverpool a few days ago (in the League Cup semi- finals).