After a 1-1 draw in regulation time, the shoot-out saw goalkeeper Courtois, an unusual choice to take his side's second penalty, bizarrely fire his attempt way over.
That was followed by new signing Morata missing the target on his competitive debut, with Olivier Giroud going on to score the winning penalty.
In the match itself, Victor Moses, sent off when the Blues lost the FA Cup final to Arsenal in May, struck early in the second half with what looked like it would be the winner.
But Chelsea had Pedro sent off for a rash challenge on Mohamed Elneny and Arsenal substitute Sead Kolasinac, on his competitive bow, scored a header from the resulting free-kick with eight minutes remaining.
Alexandre Lacazette, given his Arsenal debut from the start, had earlier come the closest to a first-half goal at Wembley, sending a strike against the post.
Arsenal have now won the Community Shield three times in the last four years, while the Blues have lost four consecutive appearances in the fixture.
Victory for Arsene Wenger's men means the Shield has seen the FA Cup winners beat the defending Premier League champions for four consecutive seasons.
Despite being without the likes of Mesut Ozil, Alexis Sanchez and Aaron Ramsey, Arsenal were the first to threaten, Danny Welbeck heading straight at Courtois.
Arsenal continued to shade the early stages and came closest to a goal in the opening half. In a fast break, Lacazette exchanged passes with Welbeck and curled his effort against the post from just inside the penalty area.
The Gunners were forced into a change around the half-hour mark, Per Mertesacker replaced by Kolasinac after drawing blood in an aerial challenge with Gary Cahill, a blow for Wenger, who was already without Shkodran Mustafi and had opted to leave out Laurent Koscielny.
That gave Chelsea some momentum, with Moses and Pedro forcing Petr Cech to parry away their efforts in quick succession.
Willian then went into the book after he was adjudged to have dived under a challenge from Hector Bellerin, before the lively Alex Iwobi showed good footwork before testing Courtois at the other end.
Chelsea made the breakthrough just 50 seconds into the second half when a corner was poorly cleared by Granit Xhaka, allowing Cahill to head back into the area, with Moses timing his run to control and beat Cech from six yards.
Wenger made two changes with 24 minutes remaining, Theo Walcott and Giroud replacing Iwobi and Lacazette, and Xhaka almost equalised soon after with a superb 30-yard strike, Courtois producing a fingertip save to deny him.
The Blues appeared to be closing in on victory as new signings Morata and Antonio Rudiger were handed late debuts from the substitutes' bench, but the match turned when Pedro was shown a straight red card for bringing down Elneny with his studs raised from behind.
From the free-kick that followed, Kolasinac rose to head into the bottom corner from Xhaka's precise delivery to send the match to penalties.
The new ABBA order for shoot-outs was used and there was a routine start with Cahill, Walcott and Nacho Monreal all scoring.
The penalties took a dramatic turn when surprise-taker Courtois, and then Morata, both missed the target, allowing Oxlade-Chamberlain and Giroud to seal victory for the Gunners.
Key Opta facts:
- This was Arsenal’s third shoot-out in the Community Shield, having lost their previous two against Manchester United in 1993 and 2003.
- The last two red cards issued in Community Shield matches have been to Chelsea players (Branislav Ivanovic in 2012, Pedro in 2017).
- There were just 82 seconds between Pedro's red card and Sead Kolasinac's equaliser.
- Kolasinac became the first player to open his goalscoring account for Arsenal in the Community Shield since Gilberto Silva in 2002.
- Victor Moses scored his first goal in 27 games for Chelsea in all competitions, since netting against Tottenham in November 2016.