Klopp’s side won 6-5 on penalties at Wembley as Greek defender Kostas Tsimikas was the unlikely hero with the decisive kick after Alisson Becker kept out Mason Mount’s effort.
A hard-fought final had finished a 0-0 draw after extra-time, setting the stage for a nerve-jangling shoot-out that featured misses by Chelsea’s Cesar Azpilicueta and Liverpool’s Sadio Mane, whose shot was saved by Edouard Mendy when he had a chance to clinch the trophy.
For the second time this season, Liverpool emerged victorious against Chelsea on penalties, having already beaten them 11-10 following another 0-0 draw in the League Cup final in February.
Liverpool’s first FA Cup triumph since 2006 keeps them in the hunt to become the first English team to win all four major trophies in a single season.
With two prizes already secured, the Reds will hope for a slip from Premier League leaders Manchester City, who are three points clear of them in the title race with two games left for both teams.
“It was an incredible, intense game. Chelsea would have deserved it exactly the same way, but that’s how small the margins are,” Klopp said.
“We overcame good moments from Chelsea, then had really good moments ourselves.
“We are mentality monsters but there were mentality monsters in blue as well. Chelsea played outstanding but in the end there must be one winner and that was us today.”
Even if Liverpool are unable to win a second English title in three seasons, they could still cap an astonishing campaign by winning the Champions League final against Real Madrid in Paris on May 28.
With that ambition in mind, Klopp’s only concern as he danced jubilantly in front of the Liverpool supporters amid a swirling mist of red flare smoke was whether his team will be at full strength against Real.
Liverpool lost Mohamed Salah and Virgil van Dijk to injuries that could rule them out of their last two Premier League games next week and will spark fears about their ability to recover in time for the Champions League final.
“I think Virgil is fine but his muscle was hurt,” Klopp said.
Liverpool have won both domestic cups in the same season for the first time since 2000-01, with Klopp getting his hands on the famous old trophy for the first time.
For Chelsea, it was a painful third successive FA Cup final defeat after previous losses to Leicester and Arsenal.
“I have no regrets. I told the team I’m proud. We struggled in the first 15 minutes, we looked nervous and tense but then we were excellent,” Chelsea boss Thomas Tuchel said.
“We held maybe the most dangerous attacking team in the world to 0-0 and created chances ourselves.
“We are sad. Everybody is in pain. We have to digest it and keep on going. That is life in sport.”
Diaz should have put Liverpool ahead during their blistering start when he accelerated onto a sublime pass from Trent Alexander-Arnold, but Mendy saved and Trevoh Chalobah scrambled back to clear off the line.
With Tuchel gesticulating furiously on the touchline, Chelsea dug in and finally turned back the red tide.
Chelsea’s Christian Pulisic clipped his shot just wide and Marcos Alonso’s poor touch squandered another good opportunity.
Chelsea were on the offensive after the interval and Pulisic’s strike was well saved by Alisson before Alonso’s free-kick cannoned off the bar.
But Klopp’s side haven’t come this close to a quadruple without remarkable fortitude.
They wrestled back the momentum and Diaz clipped the outside of the post before Robertson volleyed against the woodwork.
Tuchel had sent on his penalty-saving specialist keeper Kepa Arrizabalaga ahead of the League Cup final shoot-out in a move that backfired.
Tuchel stuck with Mendy this time, but the result was still the same as Tsimikas rolled his penalty home to trigger wild celebrations from Klopp and company.