The 38th-ranked Dutchman came through 6-7 (4/7), 7-6 (7/3), 6-3 after two hours and 37 minutes against number 103 Thompson of Australia.
Griekspoor, 26, had also won the Pune hard court title in India at the start of the year.
He is the second successive homegrown champion at the Wimbledon warm-up event after Tim van Rijthoven upset Daniil Medvedev in the 2022 final.
"I want to thank the crowd for getting me through today," said Griekspoor who will rise into the top 30 next week and guarantee a Wimbledon seeded spot.
"In finals it's so tough to play your best tennis. I had so many nerves today, I felt so tight. I wanted it so badly. He was playing great. I had to try and focus on my own game and try and hang in there."
In the women's event, Ekaterina Alexandrova came from behind to defend her title, battling past fellow Russian Veronika Kudermetova in three sets.
Alexandrova, seeded fourth, dropped the first set and conceded her first service game in the second before her fightback saw her win 4-6, 6-4, 7-6 (7/3).
Top seed Kudermetova also lost last year in the semi-finals to Alexandrova who lifted the fourth title of her career after two hours 52 minutes of play.
"It was such a joyful moment to be here again," said Alexandrova after the longest WTA final of the year.
"I just needed to really leave everything on the court. If I didn't win, great for her, she played amazing, but at least I'd know I did everything I can. In the end, it finally paid off, and it feels amazing."
Kudermetova raced to an early 4-2 lead before the match was stopped for an hour rain delay. She came back to wrap up the set and earned a break at 6-4, 3-2.
But an Alexandrova backhand return down the line for 3-3 rekindled her hopes. After another shorter rain delay at 5-4 Alexandrova came back on court and broke again to force a third set.
Alexandrova held one championship point at 5-4 and two more championship at 6-5, but Kudermetova's serve pulled her through to the decisive tiebreak.
World number 26 Alexandrova took control earning three more championship points at 6-3 with a Kudermetova backhand into the net sealing the victory.
