The Irishman scored twice from the spot after clumsy challenges from Alex Davey and Daniel Pappoe for the Blues, who also saw Alex Kiwomya dismissed late on
Burke calmly slotted in three minutes after half-time after Alex Davey had felled Callum Robinson, and added a second in injury time after Daniel Pappoe hauled down Josh Barton.
Chelsea dominated possession in the match, but Villa were well worth a win founded on disciplined defending in numbers and clinical finishing from 12 yards.
The result sees Villa become the first ever English NextGen champions, and might also provide a silver lining for the club's fans on a weekend which saw the senior side drop into the Premier League drop zone.
Jeremie Boga shifted to the left flank for Chelsea to cover for Adam Nditi, injured in the semi-final win over Arsenal, while Nathan Ake returned to anchor the midfield.
Villa also made one change, with Riccardo Calder replacing Jordan Lyden at left-back, while Michael Drennan, scorer of six goals in this season’s tournament, remained absent.
Both sides spurned chances to take the lead in a frenetic opening minute. First, a mistake from Ake allowed Jack Grealish a free strike at goal from 20 yards, but the winger blazed over.
Moments later, Lewis Baker surged through the middle before feeding Alex Kiwomya, but Bradley Watkins rushed off his line to smother the attempted shot.
Thereafter the game settled into a pattern, with Villa sitting deep and Chelsea dominating possession, though the bumpy Giuseppe Sinigaglia pitch made passing football difficult.
Kiwomya continued to be the Blues’ best outlet, and on 22 minutes Baker sent him racing through, but the pacey winger fired wide with only Watkins to beat.
Only two minutes later Boga surged into space down the right, but his low cross was intercepted by a stretching Watkins with Islam Feruz waiting to tap home.
Villa then wasted a similar chance, as Calder raced to the byeline and centred the ball low. Mitchell Beeney fumbled, but the ball got stuck under Robinson’s feet with the goal gaping.
Chelsea soon went even closer when Kiwomya found Ake in the area. The Dutchman poked the ball past Watkins but Calder cleared off the line, leaving the scores level at the break.
The first half had been one-way traffic but it was Villa who took the lead three minutes into the second half, following a mistake from Alex Davey.
Robinson tempted the young Scot into a clumsy challenge in the penalty area and, when the referee pointed to the spot, Burke nervelessly sent Beeney the wrong way.
Rattled, Chelsea continued to monopolise the ball but without the same sense of purpose, and Villa’s massed defensive ranks frustrated them with ease.
Indeed, it was Villa who now looked the more dangerous, and three minutes later Robinson fed Samir Carruthers on the edge of the area, only for the captain to blaze over.
On 69 minutes an increasingly confident Villa threatened again, when Barton unleashed a 25-yard curler which a stretching Beeney tipped over.
But it mattered not. The belief visibly drained away from a frustrated Chelsea who had Kiwomya sent off late on for a second bookable offence.
And Burke dispatched a second penalty with the last kick of the game after Pappoe had brought down Barton, as Villa closed out a famous win with ease.