Western Illinois had tied the game 21 seconds earlier.
Miller found Wooden wide open deep down the left side for the game-winner. The Leathernecks (0-3) had tied the game on Clay Bruno's 5-yard keeper after driving WIU 73 yards.
The Thunderbirds (2-1) had six takeaways with four interceptions and two fumble recoveries.
Southern Utah took advantage of a short field to score first on Micah Pettit's 27-yard field goal. The Thunderbirds made it 10-0 in the final minute of the first quarter after Julian Sanderlin's fumble recovery led to Elijah Burns' 5-yard touchdown two plays later.
Miller was 20-of-33 passing with 194 yards, a touchdown and an interception.
Bruno had 219 passing yards and 70 rushing but threw three of the picks. Naseim Brantley had eight catches for 116 yards.
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