Centennial's Terry Davis scored with seven seconds remaining in the first overtime to send the game into double overtime.
Mays and Davis both finished with a game-high 14 points as did Centennial's Malik Foxx.
Millwood (19-9) now meets Okemah at 9:15 p.m. Saturday at State Fair Arena for the state 3A title.
The Falcons will be seeking their 15th state championship in boys' basketball. Okemah will be trying to win its first.
Edwards said the Falcons will have plenty of energy left for Saturday night's title game despite the two emotional wins at the buzzer against Hugo and Centennial.
“We love games like this,” he said. “We got enough for tomorrow. We are going to bring it 10 times harder tomorrow.”