Verstappen led all but one lap around the streets of the Principality after being promoted to pole position when Charles Leclerc failed to make the grid after damage to his Ferrari was discovered from his Saturday qualifying crash.
"It's of course so special around here to win," said the Red Bull driver. "It's also for me the first time on the podium here so an amazing race.
"It's a lot of laps around here and you really have to keep your focus but it's really cool."
He added: "You never know what is going to happen but it was all about just looking after your tyres and finding a good stop gap.
"Of course, the others went early on so that made it a bit easier for me but it was pretty much in control."
Verstappen capitalised on a tough day for Mercedes and reigning champion Lewis Hamilton to leave Monaco with a four-point lead in the drivers' standings after his title rival trailed home seventh.
Although leading the drivers' standings for the first time in his six-year F1 career, the Dutchman is solely focused on the bigger picture.
"Of course, you always want to win this grand prix," added Verstappen. "I remember when I was very little already watching this grand prix.
"When I'm standing here, of course, I am very proud but I'm mostly thinking ahead.
"It's still a very long season and, of course, this is again a great way to continue."
Williams "fantastic honour" to wave chequered flag
Verstappen was waved across the line by 23-time Grand Slam winner Serena Williams, the tennis superstar taking a brief pause from her preparations for the French Open.
"It's such an amazing grand prix and such a fantastic honour to be here as a fan and see such incredible athletes," she said.
"We're all training really, really hard and it's always good to see amazing athletes and congratulations so it's always good to see and just be inspired really.
Asked if she had any advice for the new championship leader, Williams commented: "I don't think I'm in the position to give advice right now in this arena but good job to all the racers."
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