The Bulgarian took a first-set lead but Tunisia's Jaziri, ranked 117th in the world, rallied to beat the world number four 4-6 7-5 6-4.
Jaziri, 34, will now play Dutchman Robin Haase in the second round.
"You have days like this that you can't really do much else," said Dimitrov, runner-up in Rotterdam a week ago.
"Unfortunately, I couldn't play my game to the extent that I was looking for. My movement was not good over the court. I thought I served OK for a little bit, but then I lost my rhythm again."
Jaziri said it was a "great feeling" to win his first match against a top-five player, adding: "I am really happy for that. I tried to fight every point since the beginning. It was so tight."