The Dallas signal-caller, who overcame a fractured rib and a collapsed lung suffered in last week's victory over San Francisco, scrambled to his right to avoid the blitz on 3rd-and-21 and connected with Bryant in the middle of the field for a 30-yard gain.
It became a 45-yard play when a facemask penalty on cornerback DeAngelo Hall was added on.
Three plays later, Bailey, a rookie from Oklahoma State, then split the uprights with the game-winning, 40-yard boot for Dallas (2-1).
Romo finished 22-of-36 for 255 yards, no TDs and an interception.
Rex Grossman connected on 22-of-37 pass attempts for 250 yards, one touchdown and a pick-off for Washington (2-1).