Seamer Hoggard became only the 10th Englishman to take a Test hat-trick as West Indies crumbled to 94 all out in their second innings, leaving England a target of just 93.
Captain Michael Vaughan made 32 and Marcus Trescothick 42 before Mark Butcher saw England home in just 20 overs to give them a 3-0 lead in the four-match series.
It was also the first time England have ever won three Tests in a series in West Indies.
The winning runs were greeted by roars of jubilation from thousands of England fans who all but turned the Kensington Oval into their home venue.
Captain Michael Vaughan, however, seemed almost speechless before leading his players on a prolonged lap of honour in front of their chanting supporters.
"It means so much," he said. "It hasn't really sunk in. It's an amazing achievement."
"We expected a tough day when we arrived this morning."
"We'll definitely enjoy tonight and we'll probably enjoy tomorrow."
Rival skipper Brian Lara added: "We really have to buck up our batting."
"I think the guys are in a state of shock. The guys are really hurting."
While man of the match Graham Thorpe kept England in the game on Friday with an undefeated 119 out of a total of 226, it was Hoggard who put the finishing touches.
West Indies resumed on 21 for one but Hoggard ripped the heart out of the home side with four for 25 from 11.1 overs before lunch before the innings ended in 42.1 overs.
Only Lara offered any resistance with 33 but he was helpless as his side's frail batting was exposed again, just as they had been in the first Test in Jamaica when they were skittled for 47.
Strike bowler Steve Harmison took three for 34 and all rounder Andrew Flintoff, the key bowler of the first innings with five for 58, chipped in with two for 20.
West Indies lost Daren Ganga to Hoggard for 11 but Lara and vice-captain Ramnaresh Sarwan looked in the mood to battle.
But Lara, who suffered a nasty blow on his elbow from a Harmison delivery, could only watch from the non-striker's end as Hoggard took England's first hat-trick since Darren Gough against Australia in Sydney in 1999.
First Sarwan drove straight to Ashley Giles at gully. Then left-hander Shivnarine Chanderpaul was trapped lbw by a ball which nipped back and Ryan Hinds, rooted to the crease, was snapped up at second slip Flintoff.
Hoggard, who finished with four for 35, conceded afterwards there had been an element of luck to his hat-trick delivery.
"I was thinking: 'Bowl your best ball, an inswinger, on to leg stump'. It didn't really swing. It went straight across and thankfully he nicked it," Hoggard told Sky Sports.
He had barely recovered from being mobbed by his team mates after his triple-wicket maiden before it was 48 for six as wicketkeeper Ridley Jacobs gloved a Flintoff lifter to Butcher at gully.
After light rain forced an early lunch, Lara and Tino Best put on 32 for the seventh wicket, but after a second rain delay, Best fell first ball back, caught behind off Harmison for 12.
Lara was in the wars again as he suffered two more painful blows, one to each arm, but West Indies' pain worsened when Pedro Collins was run out by Nasser Hussain.
Any hope West Indies had of setting a defendable target ended when Lara was dismissed, caught at mid-on by Michael Vaughan off a mis-timed pull off Harmison.
After more light rain, Vaughan signalled England's intent with a pull over square leg for six and a sumptuous off-drive for four before he was caught by Jacobs off Corey Collymore for 32.
Trescothick fell with two runs needed but a single to Butcher, off a no ball, gave England a famous victory.
The fourth Test starts on April 10 in St John's, Antigua.
