The sides, meeting in a second domestic cup final this season, played an entertaining, scoreless draw over 120 minutes before deciding it on penalties.
It was a carbon copy of the League Cup final in February, which Liverpool won 11-10 on penalties after that game also finished goalless.
Saturday's win means Liverpool remain on course for an unprecedented quadruple of trophies.
They are three points behind Manchester City in the Premier League race with two games to go and meet Real Madrid in the Champions League final on May 28.
"I'm really proud of my boys," Liverpool manager Juergen Klopp told the BBC. "Chelsea were outstanding, but in the end there can only be one winner and today it was us."
For Chelsea, it meant an unwelcome record of becoming the first side to lose three FA Cup finals in a row, after losing to Arsenal in 2020 and Leicester City last year.
Liverpool were on top early with Luis Diaz a particular threat. A through-ball from Trent Alexander-Arnold put Diaz through on goal in the eighth minute, but Edouard Mendy denied him before Naby Keita put the rebound wide.
Chelsea responded well to Liverpool's pressure, with Christian Pulisic going close on 23 minutes after excellent play between Mount and Reece James on the right wing. Shortly after, Pulisic's pass gave Marcos Alonso a sight of goal, but Alisson was off his line quickly to deny the Chelsea defender.
Chelsea made the faster start after the break with Pulisic drawing a save from Alisson and Alonso striking the crossbar from a free kick, but both sides toiled and could create only half-chances for much of the second period.
Neither side were able to find a breakthrough in the additional 30 minutes, meaning spot kicks for the second time this season.
Mendy had kept Chelsea alive when he saved Sadio Mane's penalty, but it proved to no avail, as Liverpool won the trophy for the first time since 2006.
Sadly for Chelsea, as in February, it was a case of so near yet so far, although manager Thomas Tuchel had no regrets.
"Like in the last final, the League Cup, no regrets. I told the team I was proud," he told reporters. "We played 240 minutes in two finals against maybe the most dangerous attacking team in the world and it's 0-0.
"I was sure during the match the momentum was on our side, but unfortunately I was not right."
Liverpool captain Jordan Henderson was handed the cup by Prince William to roars from the Liverpool fans, while the Chelsea end was largely empty.
"It's a big moment for us -- we haven't been in this final for some time, so to win it was special," Henderson said.
The win was Liverpool's eighth in the 150-year history of the FA Cup, drawing them level with Chelsea and Tottenham Hotspur. Arsenal have won the most with 14.
Information from Reuters was used in this report.