The world number one downed Germany's Yannick Hanfmann 6-3, 6-3 in the second round at the Parc des Eaux-Vives grounds.
Djokovic becomes only the third man after Jimmy Connors and Roger Federer to reach the 1,100-win landmark.
Djokovic took a wild card to play in Geneva in a bid to rescue an alarming dip in form ahead of next week's French Open, the second Grand Slam of the year. His record this year is now 13 wins and five defeats, and he has yet to reach a tournament final in 2024.
After beating Hanfmann, the crowd sang "Happy Birthday" to Djokovic as a cake with candles was brought out onto the court. He lifted the cake and showed it off to the crowd, having a nibble before offering some to the ball boys and girls.
"The key was the birthday. My birthday probably wouldn't be the same if I didn't win the match," the Serb said.
"It's nice to be here for the first time in this tournament and to have my family here coming to support as well. I grew up with my aunt and my uncle and my two cousins for most of my life," he said, referring to his relatives who now live in Geneva.
"We don't get a chance to see each other that much lately so it's really nice. I'm getting some really quality time on the court but also off the court.
"Thank-you so much for singing the 'Happy Birthday' song. That really touched me so I'm very happy to win on this special day."
Ruud into last-eight
In a match interrupted by rain, Djokovic was 3-0 down in the second set but rallied to close out the contest.
Djokovic was striking the ball cleanly and consistently from the baseline and throwing in occasional drop shots. He will face either Denis Shapovalov or Tallon Griekspor in Thursday's quarter-finals.
Meanwhile second seed Casper Ruud, who won the Geneva title in 2021 and 2022, booked his spot in the quarter-finals with a 4-6, 6-2, 6-2 win over Austrian qualifier Sebastian Ofner.
It was Ruud's 32nd tour-level win of the season. No player has earned more victories in 2024.
The Norwegian world number seven - who was beaten by Djokovic in last year's French Open final - will next face Argentinian fifth seed Sebastian Baez.
The Czech Republic's Tomas Machac will face Alex Michelsen in the last eight after the American knocked out his third-seeded compatriot Taylor Fritz 6-4, 4-6, 7-5.
Michelsen only claimed his first tour-level clay-court win last month.
Thursday's other quarter-final will feature Alexandr Shevchenko against Flavio Cobolli, who knocked out US fourth seed Ben Shelton.