Bailey, who opened the scoring with a four-yard run, broke a 7-7 tie 56 seconds into the fourth quarter with a three-yard run that capped a 19-play drive.
Rice (2-4) produced 247 yards on the ground with six different players rushing for at least 20 yards. Bailey finished with 65 and Greg Henderson had a team-high 68.
The Owls converted four of four fourth-down opportunities and scored all three times in the red zone.
Kyle Herm completed six of nine passes in the first half before sustaining a concussion in the second quarter.
Navy tied the score at 7-7 with 4:16 to play in the first when Craig Condeto hit Eric Roberts for a seven-yard pass on the team's first possession of the game.
Condeto completed six of 11 passes for 53 yards and led Navy with 58 rushing yards despite receiving a right thigh bruise late in the first half.
Rice's most recent road win had come against Navy last October.
The Midshipmen (1-5) have lost five straight since winning the season opener over SMU. Navy has dropped 13 straight home contests.