The result at the Amex Stadium helped Brighton end a run of nine straight league defeats by Chelsea stretching back to 1983 while Frank Lampard's side missed the chance to seal back-to-back league wins for the first time since November.
Azpilicueta celebrated his 100th start as Chelsea captain with the opening goal, guiding the ball home from close range in the 10th minute after Tammy Abraham's shot was blocked near the goal line by Brighton's Aaron Mooy following a corner.
Substitute Jahanbakhsh then produced an early contender for goal of the year in the 84th minute, with a superb overhead kick that flew into the bottom left corner giving Chelsea goalkeeper Kepa Arrizabalaga no chance.
"It is an amazing feeling. When I was about to go on, I had a feeling I could have an impact on the game," Jahanbakhsh, who scored his first Brighton goal in last week's 2-0 victory over Bournemouth, told the BBC.
"When I saw the ball, I just tried to hit it as hard as I could and the bicycle kick was the only option. I'm going to watch it a couple more times.
"I'm very glad to help the team win points. I couldn't start the year any better... we did a great job."
Arrizabalaga had earlier kept out substitute Aaron Connolly's strike from close range and came to Chelsea's rescue again after the equaliser when he saved Neal Maupay's first-time shot from Martin Montoya's cross.
Chelsea also had opportunities to take all three points with their best chance falling to substitute Callum Hudson-Odoi, but his curled effort from distance narrowly missed the target in the 88th minute.
Lampard said he was frustrated by Chelsea's inability to kill off the game in the opening period.
"In the first half, the game was there to be won. We weren't ruthless enough. We got the goal and we had the possession. But we just didn't kill the game off," Lampard said.
"We allowed them to stay in the game. In the end it was a wonder goal but we were lucky not to lose. The tactics matched up for both teams but they were better than us. We have to take that on the chin."