The Italian, 60, was sent to the stands by referee Kevin Friend in stoppage time after losing his cool during the closing moments of a heated clash at Stamford Bridge.
The FA confirmed the sanction in a statement on Thursday.
Sarri's assistant Gianfranco Zola said Burnley's backroom staff had "offended" Sarri while Chelsea's David Luiz and Burnley's Ashley Barnes were involved in a scuffle after the final whistle at Stamford Bridge.
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Afterwards Luiz accused Burnley of "anti-football".
(Reporting by Martyn Herman; Editing by Toby Davis)
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