The Cowboys also captured the NFC East title with the win.
Witten had a career-high 15 catches for 143 yards after catching the winning touchdown with 18 seconds left in the game, while Tony Romo threw for 307 yards on 35-of-44 passing with two touchdowns.
The Cowboys (12-1) have now won their seventh straight game.
Jon Kitna managed just 22 out of 36 passes for 248 yards for Detroit (6-7), which lost for the fifth time in a row.
18-year-old elf alleges mall Santa pinched her buttocks on the job
Theater accidentally screens 'Nymphomaniac' trailer instead of Disney's 'Frozen'