The Gunners have now won nine straight matches at Wembley - including the FA Cup final 71 days ago.
Kolasinac's leaping header in the 82nd minute cancelled out Victor Moses' advantageous opener just after the interval and ensured the August showcase would go to penalties.
It gave spectators the opportunity to witness the new ABBA format, which the English Football Association adopted in order to give the team shooting second a better chance of winning.
And it worked. Chelsea's Gary Cahill converted the first kick, but that was all Antonio Conte's side could muster.
The decision to let Courtois take Chelsea's second penalty backfired, as the Belgian's attempt soared over the crossbar. Goalkeeper Petr Cech extended to deny new-boy Morata seconds later.
It was a deserved victory for Arsene Wenger's men, who, despite the club-record-signing of Alexandre Lacazette, have entered the season with tempered expectations.
Arsenal dominated the opening few minutes and could've grabbed an early lead had Lacazette not pinged the upright with a devastating curler.
Chelsea slowly grew into the match, but continued to trail its counterpart in chances created and possession.
Things changed after half-time. Barely a minute into the second stanza, Moses broke free of his markers in the box to slip a shot past Cech.
From the subsequent free-kick, Granit Xhaka, who produced quality set-pieces all match, delivered the cross that Kolasinac nodded in.