Raheem Sterling smashed home the decisive spot kick as City clinched the English season's first silverware 4-3 on penalties, with the Wembley showpiece having ended 0-0 after extra time.
The clash, the first between the clubs in a major final, had slowly warmed up, but took the most unlikely of twists deep into the extra period when Arrizabalaga saved from Sergio Aguero.
It appeared the young Spanish keeper had injured himself and Sarri immediately told substitute Willy Caballero to get ready to come on for the closing seconds as penalties loomed.
With his number being held up, Arrizabalaga refused to obey the call to depart, leaving Sarri fuming in the technical area.
At one point the raging Italian coach appeared to walk down the tunnel, before returning to brief his players for penalties.
Caballero still looked poised to take over but it was Arrizabalaga who stood guard in front of the Chelsea fans.
The Spaniard allowed Sergio Aguero's weak penalty to creep under his body into the net but then redeemed himself with a brilliant save from Leroy Sane leaving the shootout at 2-2.
Chelsea's David Luiz thumped his penalty against the post, however, to hand the advantage back to the Manchester side and, although Hazard scored, Sterling smashed his kick high past Arrizabalaga to send City's fans into delirium.
UNDER PRESSURE
Sarri began the match under pressure with his inconsistent side having lost five of their last 11 games, including a 6-0 thrashing at City in the Premier League and a 2-0 home loss to Manchester United in the FA Cup fifth round.
The United defeat provoked fury from Chelsea fans who have lost patience with the former Napoli manager's tactics.
A fortnight after the humiliation by Pep Guardiola's side's, Sarri's tactics were spot on as Chelsea stifled City's much-vaunted attack and created the better chances.
Had N'Golo Kante not blazed over after being teed up by Hazard's brilliant run midway through the second half, the headlines could have been about Sarri winning his first silverware after 29 years as a manager.
Instead, he was left answering questions about what looked like a blatant case of a player undermining his authority.
"I misunderstood the problem and only realised the situation when the doctor arrived at the bench," he said later.
"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."
HARD WORK
City have now won the League Cup four times in six seasons and are second on the all-time list, with six titles, behind title rivals Liverpool who have won eight.
They had to work hard for it, though, with Guardiola correctly predicting before the match that Chelsea would be desperate to avenge their demolition at the Etihad.
"Chelsea have incredible quality and after the result two weeks ago we knew it would be more difficult," said Guardiola, who picked up his third major trophy for City and the 25th of a managerial career spanning three countries.
Asked if his side could claim four trophies this season, and surpass the treble he achieved with Barcelona in 2008-09, Guardiola added: "What we did in Barcelona is unique. This is new players, new club, now we will see how far we can go."
City dominated possession in the first half but only had a couple of Aguero attempts to show for it.
Chelsea were rejuvenated after that with Hazard coming to the fore against an increasingly uneasy City side.
Aguero did have a goal ruled offside and also had chances in extra time, but Chelsea had looked the more likely winners.
(Reporting by Martyn Herman; Editing by Ken Ferris)
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