The summer arrival from Bologna was an unused substitute in the Community Shield against Manchester City last weekend, but he was introduced in the 61st minute of Friday's contest with Fulham.
Zirkzee proved to be the hero for his new club in the 87th minute, as he turned a low Alejandro Garnacho cross into the bottom corner from inside the box.
The 23-year-old is the 15th Dutch player to score on his Premier League debut, while he is the fourth to achieve that at Man United after Ruud van Nistelrooy, Alexander Buttner and Donny Van de Beek.
Van Nistelrooy, who is viewed as one of the team's best-ever strikers, is now back at the club as an assistant to manager Erik ten Hag.
Zirkzee opens up on "perfect" Man United debut
"It's the perfect one, no? In front of the Stretford End. And a win as well, it couldn't be any better," Zirkzee told BBC Sport after the match.
"It's been very good [since I arrived]. The group have been amazing, I feel very comfortable and at home here. I'm having a very good time.
"Of course it's a big club, but we just work hard every day and focus on what we want to achieve, and that's being the best we can be. At the moment I don't really feel any pressure.
"I look forward to working with everyone here. That includes all the staff. Everyone has been great. Of course having [Ruud] Van Nistlerooy in the staff as a striker, you have to be grateful for that. Everyone has been great, it's a pleasure to be here."
Ten Hag hails Zirkzee for scoring on first appearance
Zirkzee showed his strong link-up play during his time on the field on Friday, but head coach Ten Hag said it was "very important" for the striker to find the back of the net.
"It was very important to get the first goal as a striker. For him it is very pleasing because strikers have to score. That is the first ability you have to show when you come as a striker to a new club," the manager told BBC Sport.
Zirkzee arrived at Man United off the back of scoring 14 goals and registering nine assists in 58 appearances for Bologna.
Ten Hag discusses Zirkzee's threat
"It doesn't matter if he's Dutch, English or Russian or South American, he is a Manchester United player. It is very good for a striker to come into Manchester United and to score the winning goal in his first game," Ten Hag added in his post-match press conference.
"Remember last season, when Rasmus Hojlund in his first Old Trafford Premier League game, scored a goal and we're all waiting on the halfway line with VAR and after two minutes they came to the conclusion the ball was over the line.
"It took too long and we were so frustrated. We are very happy with him, it is very pleasing and we are very happy with him. It is also encouraging and it will give him courage and belief - and for the fans. It is a very good start.
"He has some attributes we didn't have, straight away he showed that. He is very good in his linking up and combinations and we have some good ball-playing offensive players and they can take benefit from this.
"But that is what I wanted to say, he has to arrive in the box. He has to arrive in the box and score goals, he did and the winner is very pleasing for everyone."
The forward was in the Netherlands squad for this summer's European Championship, but he was only a late substitute in two matches for his country.
Zirkzee could make his full debut for Man United when the 20-time English champions head to Brighton & Hove Albion in the Premier League next weekend.