Aubameyang cancelled out Christian Pulisic's smart opener from the spot before a brilliant chip clinched the first trophy of Mikel Arteta's managerial career.
Mateo Kovacic was shown a soft second yellow shortly after Aubameyang's second to leave Chelsea with a mountain to climb and Arsenal saw out the game against their depleted opponents.
Arteta decided to keep Ainsley Maitland-Niles in his starting line-up as the Arsenal coach reverted to a back three, having used a four-man defence against Watford.
Hector Bellerin also returned to the squad following a calf problem, but there was no sign of Mesut Ozil and Matteo Guendouzi in the Gunners' travelling party.
Frank Lampard was able to welcome back N'Golo Kante into the squad and opted to keep Olivier Giroud in attack to face his former club, but Willian did not recover in time to make the 20-man squad.
Rarely has there been so much at stake for Arsenal, with a 14th FA Cup on the line but also their participation in European competition next season. For Chelsea, Lampard could become the club's first English manager to win a trophy for 49 years to follow their top-four finish.
As the team in better form, it was no surprise to see Chelsea begin on the front foot with Mason Mount testing Emiliano Martinez with a strike from 20 yards.
Arsenal didn't take heed of the warning sign and found themselves a goal down just a minute later. Jorginho threaded a pass to Pulisic, who carried the ball and offloaded to Mount. His return pass found Giroud and with a deft flick into his path, the American finished calmly past Martinez.
Pulisic was really enjoying himself especially when carrying the ball through the middle and nearly got a second inside 10 minutes, but his powerful effort was saved by Martinez this time.
Arsenal responded well to going behind and started to take hold of possession and thought they had got the leveller when Nicolas Pepe curled home from 20 yards. But the flag had already gone up, correctly, for offside.
But Chelsea's high line was opening the door for Arsenal to play over the top and that's exactly how the equaliser came about. Kieran Tierney pinged a ball over the top and Aubameyang raced past Cesar Azpilicueta, who appeared to pull back the Gabon striker inside the box. Referee Anthony Taylor pointed to the spot and Aubameyang sunk the spot-kick into the corner.
Arsenal took confidence from the goal and put pressure on Chelsea, with a succession of corners before Azpilicueta was forced off with a hamstring injury.
After the break, Chelsea very nearly retook the lead through Pulisic, racing past two defenders and missing the far post by a whisker. More bad news followed for Chelsea as the American pulled up with a hamstring problem and was replaced by Pedro.
Opportunities were few and far between in the second half until Bellerin made a surging run down the right and collided with Andreas Christensen. Play continued and Nicolas Pepe found Aubameyang, who beat Kurt Zouma with some sublime skill before lifting it over Willy Caballero. It was a fine goal and a worthy winner of any FA Cup final.
Chelsea's hopes of a comeback looked to have been dashed when Kovacic was sent off for a second bookable offence, although the replay showed the referee made the wrong decision to show him a belated red card.
Emotions boiled over on the touchline when Antonio Rudiger clashed with Arteta, but Arsenal kept their composure to seal a 14th FA Cup success and their first piece of silverware under the Spaniard.