Zverev beat Griekspoor 6-4, 7-6 in Vienna to become the 30th player officially confirmed as world No. 1. The German inherits the ranking from Jannik Sinner, whose 95-week reign will come to an end next Monday.
"It's very special," Zverev said in the post-match interview.
"It's different to winning a tournament or title, but it's such a special feeling to know that the number 1 is going to be next to your name when the new rankings come out. I'm super happy and super thrilled."
The two-time Grand Slam winner disposed of Griekspoor in their first meeting since the Dutchman's comments about the US Open final, which he called "mediocre" before saying Zverev would "absolutely not" overtake Sinner as world number 1.
Zverev won the French Open and US Open this year, but didn't play Sinner or Carlos Alcaraz to win those titles as both suffered with injury issues this year. The highest-ranked player Zverev beat en route to the titles was Ben Shelton, who was ranked 10th at the time of the US Open final.
'I had to wait'
The preparation for Thursday's match was different to normal, Zverev admitted: "It felt different. I was super focused and motivated for multiple reasons, but the number one was definitely a big subject.
"I think I played a very good match, and I'm very happy about that."
Zverev, who becomes the first German world No. 1 since Boris Becker in 1991, was close to taking the ATP throne in 2022, but had to give up the fight after suffering a torrid ankle injury at the French Open.
"I had to wait, but sometimes you have to wait. It is how it is," said Zverev, who improved his 2026 record to 62-16.
"I have no words for this year - I'm just thankful, happy, and all of the above."
In the quarter-finals of the Erste Bank Open in Vienna, Zverev will face Arthur Fery.