Nothing could separate the sides as the match finished 0-0 after extra time but it was City who claimed the season's first major silverware as they won the shootout.
Raheem Sterling fired the decisive spot kick past Spaniard Arrizabalaga to complete the first part of what could be a quadruple for Pep Guardiola's side.
Arrizabalaga minutes earlier been involved in surely one of the strangest incidents in a Wembley final when he defied Chelsea manager Maurizio Sarri's orders to come off in the closing minutes of extra time after appearing to be injured.
Sarri, who had told substitute keeper Willy Caballero to take over from Arrizabalaga, was left fuming in the technical area at the open show of mutiny by one of his players.
"It was a big misunderstanding because I understood the keeper had cramp and was unable to go to the penalties. But it was not cramp and he could go to the penalties," Sarri said.
To compound matters Arrizabalaga then dived over Sergio Aguero's weak penalty during the shootout.
He did save from Leroy Sane but when Chelsea's David Luiz struck the post with his spot kick, it was left to Sterling to smash his penalty high into the roof of the net and send the City fans into delirium after a tense afternoon.
Guardiola happy
"Of course I'm happy. Compliments to Chelsea, they played an incredible performance," Guardiola, who claimed his 25th major trophy as a manager, said.
"I'm happy to win back-to-back in this competition for the first time in the club's history."
Two weeks after Chelsea were thrashed 6-0 by City in the Premier League and six days after they went out of the FA Cup at home to Manchester United, during which Blues fans heckled under-pressure Sarri, it seemed the Italian had played the perfect tactical game to subdue Guardiola's City slickers.
After a dour first half in which Chelsea stifled City, Sarri's side were arguably the better of the two as Eden Hazard was released to cause havoc in the City defence.
N'Golo Kante missed a golden chance for the Londoners in the 65th minute after dazzling work by Hazard.
Aguero had seen an effort chalked off for offside and it was his shot at the end of extra time, saved by Arrizabalaga, that triggered the bizarre events for which a slow-burner of a final will be best remembered.
City have now won the League Cup four times in six seasons to move to second on the all-time list with six titles, two behind the record eight of Liverpool.
