Record signing Morata - a late substitute - dragged a poor effort wide moments after goalkeeper Thibaut Courtois had blazed over in a first use of a new penalty shootout system.
Premier League champions Chelsea had gone ahead through Victor Moses, before Pedro was sent off for an ugly lunge on Mohamed Elneny.
From the resulting free-kick, debutant Sead Kolasinac headed the Gunners level to take the match to penalties.
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