The striker, whose future with the Gunners remains uncertain, slotted in a first-half penalty to cancel out Christian Pulisics opener, before firing home a classy match-winning strike in the 67th minute.
It meant the battle of the rookie bosses was won by Mikel Arteta, with Frank Lampards Chelsea ending the game a man down following a red card for Mateo Kovacic.
Arsenals success means they earn a place in next seasons Europa League, having finished a disappointing eighth in the Premier League.
An early breakthrough came after five minutes for Chelsea when Pulisic clipped in a neat close-range finish after former Arsenal battering ram Olivier Giroud cleverly eased the ball into his path. It made Pulisic the first American to score in an FA Cup final, but his match would later end in injury agony. Arsenal briefly thought they were level when Nicolas Pepe thrashed a fine 20-yard strike past Willy Caballero.
An offside flag saved Chelsea that time, but moments later, in the 26th minute, Arsenal had a penalty after Cesar Azpilicueta dragged down Aubameyang. Aubameyang lashed the ball into the bottom right corner, and Chelsea captain Azpilicueta suffered a further blow five minutes later when an apparent hamstring injury ended his involvement.
Pulisic darted through in the first minute of the second half, racing from his own half to the edge of the Arsenal penalty area.
Glory was again in his sights but suddenly horror entered the Pennsylvanians eyes, a leg injury striking just as he shaped to shoot. As well as missing the target, Pulisics race was run, Pedro coming off the bench. Where Pulisic had been Chelseas man most likely, there was no doubting who Arsenal were looking to, and Aubameyang delivered a delicious strike in the 67th minute.
Hector Bellerin charged out of defence and Pepe picked up possession, feeding Aubameyang, who had plenty of work left to do.
The former Borussia Dortmund man made it look ridiculously easy as he zig-zagged past Kurt Zouma and floated a sweet finish over the diving Caballero.
Chelsea were in all sorts of trouble moments later when Kovacic received his second yellow card and subsequent red after he and Dani Ceballos battled for the ball, their challenge fading from that point. Pedro was carried off in stoppage time to deal Chelsea another blow ahead of their Champions League clash with Bayern Munich next weekend.