"We've got a great staff here, add that to professional players and it's their reward not mine," Dowie told Sky Sports News.
"We're delighted to have it, but we're not certainly getting carried away."
Ipswich Town manager Jim Magilton was runner-up, while Blackpool manager Simon Grayson was third.
Dennis Wise took the honours in League One, with Morecambe boss Sammy McIlroy the League Two manager of the month.