The world No.1, who's accepted a late wildcard to compete at the ATP 250 tournament in Switzerland after a less than ideal run-up to his title defence at next week's French Open, was stopped in his tracks by rain as he attempted to serve out the first set on Wednesday.
But the Serb returned to court following the hour-long break to claim the first stanza with two aces in a row.
Perhaps still showing signs of his vulnerability after his recent struggles, Djokovic, who saved seven break points during the opening set, also dropped serve early in the second to trail 3-0.
But the 24-time grand slam champion eventually recovered to reel off six games in a row to seal his 1100th tour-level match win.
The Serb is only the third man to achieve this milestone behind Jimmy Connors' 1274 wins and Roger Federer's 1251.
"The key to win was the birthday, it probably wouldn't be the same if I didn't win," said Djokovic, after organisers gave him his birthday cake.
"It's nice to be here in this tournament for the first time, with my family here coming to support. I'm getting some quality time on and off the court.
"Grateful to all the people that came today and thank you for singing happy birthday - that really touched me."
Djokovic requested the Geneva wildcard after he was hammered in Rome in the last 32 by little-known journeyman Alejandro Tabilo two days after a freak accident in which he was hit on the head by a fan's water bottle while signing autographs.
