The Saints moved to as high as second in the league standings last month, and Mourinho has revealed that he is pleased for the Dutchman who only moved to St Mary's Stadium in the summer.
The 51-year-old told reporters: "Ronald Koeman deserves the award. He's a great guy and a good friend.
"The second reason that I'm happy is that the Manager of the Month always loses the next match. Swansea boss Garry Monk lost against us."
Southampton will hope to continue their good form at home to Sunderland on Saturday afternoon.