The 24-year-old from Christchurch dominated the six-woman final, proving untouchable over the closing lap as she added to the silver medal she won in the same event at the previous Games in Tokyo.
After advancing through the semi-finals, Andrews stayed as calm as possible in the lead-up to the final and on the start line.
"I like to try to not get myself too overwhelmed, or too over-worked, before the race.
"I was just thinking about the simple things and for me it was just another round - and I just needed to qualify, hopefully first this time!
"And I managed to cross the line first."
Andrews hit the front on the penultimate lap and, despite the challengers lining up behind her, she could not be caught, winning by 0.062 seconds from Dutch rider Hetty van de Wouw. Great Britain's Emma Finucane took bronze.
The race went "absolutely" to plan, Andrews said after her victory.
"I went into that race mentally preparing for it like it was just another race.
"I wasn't letting myself get too overwhelmed by the fact it was an Olympic final
"I just wanted to go out there and execute the best that I could, to my strengths, and use the track, so yes it did play out how I wanted."
It was an "amazing feeling" to be in first place on the final bell, she said.
"I knew it would be a challenge at that point because I could feel the other riders on my hip, but I knew that I had to give it 100 percent.
"That was my moment that I decided I'm emptying the tank and I'm not leaving anything out there."
She told Sky Sport she was honoured to become just the second New Zealand Olympic cycling gold medallist after Sarah Ulmer, who won the women's individual pursuit at the 2004 Athens Games.
"It will take a long time to process and a long time to sink in. Sarah Ulmer is an amazing bike rider, she's an amazing person so to be beside her is incredible," Andrews said.
"I think it's surreal. I tend to take a while to let things soak in, so right now it's just surreal.
"I'm very sore, I'm very hot and I'm very puffed."
She said having her father Jon Andrews - a former New Zealand track cycling representative - as her coach made victory all the more sweet.
"I think that's special. Most of the time your parents are on the other side of the fence, so to have one trackside, on this side of the fence with me, is pretty unusual I'd say for the sport, but very special."
It was a second medal of the Games for Andrews, who was part of the silver-medal winning team sprint on the opening night of track cycling competition in Versailles.
It adds to a career tally that includes a world championship gold in Glasgow last year, along with a trio of golds on the track at the Birmingham Commonwealth Games.
