Zirkzee came off the bench in the second half at Old Trafford on Friday and delivered a memorable first impression as his 87th-minute strike gave United the perfect start to the new season.
The Netherlands striker completed his switch from Bologna a month ago, but was among several United players to get an extended break after Euro 2024.
He missed all of United's friendlies and was an unused substitute in last weekend's Community Shield loss to Manchester City.
He finally made his first appearance in United red against the Cottagers, with his winner making it a moment he will never forget.
"A great feeling, it couldn't be any better. What can I say? Debut, winning, scoring in front of the Stretford End, so yeah, it's unbelievable," he said.
"I mean yeah, first game, first goal, it's always a relief to have that off your shoulders, but it goes on and we're just going to work for the next game."
Zirkzee jumped at the chance to join Erik ten Hag's side after shining in Bologna's outstanding Serie A campaign last term.
Asked why he wanted to join United, he laughed and said: "Why Manchester United?
"Some people might be crazy, but, for me, I mean this is the biggest club in the world, so it wasn't a difficult choice to make to be honest."
Zirkzee will be able to hone his craft under former United and Netherlands striker Ruud van Nistelrooy, who returned to Old Trafford as a member of Ten Hag's coaching staff in the summer.
"Yeah, it's a privilege to have someone like that," Zirkzee said.
"As a striker, it's not every day you have someone that you grew up watching, so yeah, that's pretty much it, but everybody in the staff up to now has been great and I enjoy working with these people a lot.
"It's not just him, but obviously it's an extra privilege."
