A goal in each half from Brazilian midfielder Ramires and veteran striker Drogba clinched a narrowly deserved win for Chelsea, with substitute Andy Carroll scoring for Liverpool.
The win saw Chelsea skipper Terry enter the record books as the first man to captain one side to four victories in the competition, having already lifted the famous trophy in 2007, 2009 and 2010.
"It's fantastic, it's what we live for," a jubilant Terry told the BBC.
"We spoke about winning yesterday and we've had a tough competition in the FA Cup this season and we've won. It was a great performance.
"At times people have slated us as too old, past it or not together but as a team we come together and unite when our backs are against the wall and we've done that superbly in the last 16 or 17 games."
Terry's landmark was also matched by another milestone for Drogba, whose second-half effort made him the first man to score in four FA Cup finals.
The defeat denied Liverpool the opportunity to end a largely disappointing season with a second piece of silverware following their League Cup triumph over Cardiff City in February.
However, after controlling much of the game, Champions League finalists Chelsea were made to work hard for their win, and Carroll was denied what would have been a dramatic 82nd-minute equaliser when his close-range header was parried onto the underside of the crossbar by Petr Cech.
Carroll wheeled away to celebrate, thinking the ball, had crossed the line but referee Phil Dowd and his assistants remained unmoved.
Liverpool manager Kenny Dalglish was pleased with his side's second-half fightback but admitted his team had paid for defensive naivety.
"We were excellent in the last half hour but a game of football lasts for 90 minutes," Dalglish said. "We left ourselves too much to do. You can't lose two goals in the fashion that we lost them and expect to win."
Dalglish said he initially believed Carroll had equalised but did not dwell on the issue afterwards.
"I thought it went in but if it hasn't then give the officials the credit they deserve," he said.
Chelsea dominated a poor Liverpool side throughout a poor first half that saw the Merseysiders struggle to get into the game.
Liverpool midfielders Jay Spearing and Jordan Henderson were largely anonymous, all too often leaving captain Steven Gerrard overwhelmed.
Chelsea, meanwhile, looked comfortable for most of the match, with Frank Lampard and Jon Obi Mikel screening the back four against an impotent Liverpool attack.
Drogba had been the first to get a shot on goal, hooking a volley just over the bar after only 33 seconds as Chelsea started brightly.
The Blues then took the lead on 11 minutes through a Ramires strike that owed everything to poor play from Liverpool
Spearing squandered possession in midfield and Juan Mata gratefully scooped up the loose ball and fired a pass in behind Jose Enrique for Ramires.
The rangy Brazilian shrugged off his man with ease and then took advantage of shoddy goalkeeping from Pepe Reina to lash home at the near post.
Liverpool might have got back into the match almost immediately, only for Branislav Ivanovic to thwart Craig Bellamy's fierce volley on 14 minutes with a gutsy block.
Yet that was as good as it got for Liverpool for the remainder of the half, with Cech having little to do in the Chelsea goal.
Liverpool started the second half more purposefully, with Gerrard producing one surging run into the box and Stewart Downing looking lively down the flank.
But Chelsea delivered a hammer blow on 52 minutes when Drogba pounced, rifling home a low left-foot shot after being picked out in space by Lampard.
Liverpool responded immediately by bringing on Carroll for the ineffectual Spearing but it was Chelsea who continued to look the more threatening of the two sides.
Drogba lashed a volley into the side netting but then the momentum shifted dramatically when Carroll pulled one back for Liverpool.
The £35 million ($56.5m) striker pounced after Jose Bosingwa's clearance ricocheted into his path, wrong-footing Terry to fire high into the net.
It was all Liverpool thereafter, with the Reds galvanised by the presence of Carroll in the front line.
Luis Suarez forced a low save from Cech on 73 minutes before Carroll headed just over three minutes later.
Carroll was then denied his headed equaliser eight minutes from time but although Liverpool pressed they were unable to find a way through a resolute Chelsea rearguard action.
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