The Dolphins came into the game not having allowed a touchdown in 12 consecutive quarters and nearly stretched to 14. With the score tied at 3-3, the Cowboys got a huge break as Moore fumbled a snap for the second time.
"We had an issue with that tonight," Moore said of the fumbled snaps. "I can't put a finger on why."
DeMarcus Ware recovered the loose ball on the Miami 5. Romo connected with Robinson connected on a next play, a strike over the middle with 55 seconds left.
The Dolphins did close the gap thanks to a nice return by Clyde Gates out near midfield and pass interference penalty on the Cowboys. Graham nailed a 28-yard field goal as time expired.
The Cowboys didn't get on the board until the less than five minutes were left in the first half. Murray got the drive started before it fizzled with Romo taking a couple shots into the end zone.
Bailey drilled a 32-yarder to tie the game 3-3.
Romo came into the game without an interception in his previous 126 passes -- the second longest such stretch of his career. He threw two picks in the first quarter alone, but it only cost the Cowboys three points.
"It's just about winning," Romo said.
The first turnover came inside the Miami 20 and ended a promising opening drive for Dallas. Romo's second bad throw set up the Dolphins at the Cowboys' 26. Graham cashed the field position in with a 26-yard field goal.
Graham, kicking in place of an injured Dan Carpenter, missed a 47-yarder early in the second quarter.
Game notes: Cowboys WR Miles Austin (hamstring) missed his fifth straight game. Dallas was also without FB Tony Fiammetta (illness) and CB Mike Jenkins (hamstring). ... Miami inactives included PK Dan Carpenter and NT Igor Olshansky. ... The Cowboys are 28-15-1 all-time on Thanksgiving. ... The Cowboys and Dolphins met an in infamous Thanksgiving game 18 years ago won by Miami on a last-second field goal that was set up when Dallas defensive lineman Leon Lett tried to recover a blocked field goal in the snow.