Preston, who had lost in their previous nine play-off campaigns, went ahead when Beckford struck from close range.
Paul Huntington scored with a cushioned volley from a Paul Gallagher cross and Beckford curled in from the edge of the area to make it 3-0 at the break.
Beckford's third came as he slotted in when one-on-one with Wes Foderingham.
Preston recovered from missing out on promotion on the final day of the campaign to return to the second tier after a four-year absence.
Beckford, who has been on loan at Preston but was released by parent club Bolton last week, won the free-kick that Gallagher subsequently sent into his path inside the six-yard box to give the Lilywhites the perfect start.
As well as conceding, Swindon lost captain Nathan Thompson with a leg injury as he tried - and failed - to challenge Beckford in the build-up to the goal.
Preston centre-back Huntington added a second when he arrived unmarked to tap home from Gallagher's inswinging delivery from the left - his ninth goal of the season.
Mark Cooper's Swindon side dominated possession for the majority of the game and striker Michael Smith should have got a goal back when he headed wide from Nathan Byrne's cross.
Moments later, Beckford, who scored three goals in the play-off semi-final, reached Joe Garner's flicked pass before beautifully curling a left-footed shot into the bottom corner.
He completed his hat-trick when strike partner Garner sent him clear of the defence and Beckford calmly sent the ball beyond Foderingham.
A long-range Jordan Turnbull effort and a Jonathan Obika header, which was tipped wide late on, were among the Robins' best chances to score.
Swindon boss Mark Cooper:
"We didn't deal with Joe Garner and Jermaine Beckford.
"If a ball comes in your box and you don't want to head it or compete then you're in trouble.
"You've got to give credit to Preston, they were excellent.
"Over the season we've been very, very good and entertained a lot of people."
Jermaine Beckford scored a hat-trick to help Preston end their dismal play-off record by thrashing Swindon in the League One final at Wembley.