Romo's fourth-down pass with 1:16 remaining came after Murray had been knocked back for a 9-yard loss on the previous play, setting up a do-or-die situation for the Cowboys.
The final play was set up by a 51-yard bomb from Romo to Terrance Williams earlier in the decisive drive.
Romo finished 17-of-27 through the air for 226 yards, two touchdowns and an interception for Dallas (8-7), which will meet NFC East rivals Philadelphia next week for a winner-take-all regular season finale.
Murray also logged a rushing touchdown, picking up 96 yards on 22 carries.
Kirk Cousins completed 21-of-36 passes for 197 yards, a TD and a pick-off for the Redskins (3-12).
Pierre Garcon made 11 catches for 144 yards with a score.