After goals from Victor Moses and Sead Kolasinac led to the game ending 1-1 after normal time, Thibaut Courtois and Alvaro Morata missed from 12 yards in the resulting shootout to give Arsenal a win they probably deserved.
Two bookings in quick succession for Cesar Azpilicueta and Hector Bellerin highlighted that neither side were treating the game as a friendly, but chances were few and far between until Arsenal went close midway through the half.
The ball was worked to Alexandre Lacazette just inside the penalty area and although his curling effort beat Courtois, it struck the post and went clear.
There was a break in play after Per Mertesacker sustained a nasty cut above his eye after being accidentally elbowed by Gary Cahill, but it benefited Chelsea who then enjoyed their best period of the opening 45 minutes.
A minute before the break, Alex Iwobi curled the ball straight at Courtois but both teams deserved to go into the changing rooms on level terms after a competitive half.
Straight after the restart, however, Chelsea took the lead through Moses, who converted from close range after reacting quickest to Cahill's header back into the penalty area.
Moments later, Michy Batshuayi miscued a volley straight at Cech as the Premier League champions looked to seize control, but just before the hour mark, Courtois was forced into a brilliant intervention to keep out Mohamed Elneny's driven cross.
Both teams began to make substitutions and that hindered the flow of the game, but Arsenal almost found an equaliser out of nowhere with 15 minutes remaining when the fingertips of Courtois tipped a 30-yard strike from Xhaka wide of the post.
However, after Pedro was sent off for a studs-up challenge on Elneny, Arsenal found their leveller from the resulting free kick as Kolasinac headed into the bottom corner from Xhaka's pass.
Arsenal had the momentum throughout the closing stages, but they were unable to force more work out of Courtois and both teams were forced to settle for a penalty shootout.
Using the new 'ABBA' method, Arsenal scored each of their four attempts, with Courtois drilling a spot kick over the crossbar and Morata striking the outside of the post to gift Arsene Wenger's side the win.