Their record run was halted in a 2-0 defeat to Tottenham Hotspur last week.
Chelsea's impressive form under Conte has catapulted the London club to the top of the table with 49 points from 20 games, five more than second-placed Liverpool.
"This award shows we're working very well," said Conte, who fended off competition from Liverpool counterpart Juergen Klopp, Manchester United's Jose Mourinho and Tottenham Hotspur's Mauricio Pochettino.
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"December was a great month for us. It was important for the table and for our confidence."
Manchester United striker Zlatan Ibrahimovic was named Player of the Month, having scored five league goals in December.
(Reporting by Hardik Vyas in Bengaluru; editing by Toby Davis)
