German centre-back Antonio Rudiger had powered home a header with less than six minutes on the clock for what looked like being the game's only goal, but the Blues were pegged back at the death and suffered defeat under Maurizio Sarri.
Here are the highlights as the match went to penalties and the Gunners survived a tense shootout to secure the win.
Arsenal's Alexandre Lacazette celebrates scoring their first goal with teammates.
Yet while Hudson-Odoi's rich talent offers hope for a bright Chelsea future, Sarri's most pressing task is to navigate the dysfunction that overshadows their present.More news: Trump administration confirms plan to raise China import tariff to 25 percent
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Especially with Chelsea competing in the Europa League, there should be plenty of opportunities for Hudson-Odoi to get minutes under his belt this coming season.
Arsenal improved after the interval but purposeless passing from Emery's team did not threaten Chelsea, an unmarked Rudiger heading wide from Pedro's corner as Sarri's men sought a decisive second.
Alex Iwobi's introduction on 65 minutes brought fresh urgency to Arsenal, and the Nigeria global had shots saved by both by Caballero and his replacement Marcin Bulka. Morata stepped up but telegraphed his kick and Cech dived low to his right to smother it.
In additional time, Bulka could stop Lacazatte's rebound shot from close range.