But the Cowboys couldn't score on their first overtime possession, giving Hartley the opportunity to pull out the victory for the Saints (7-8), who logged their second straight win.
The Saints' Darren Sproles contributed 101 receiving yards on seven catches as well as 48 rushing yards on nine carries.
Romo went 26-of-43 through the air for 416 yards and three scores for Dallas (8-7), which fell out of first place in the NFC East with the loss.
Dez Bryant led the Cowboys' receiving corps with nine catches for 224 yards and a pair of touchdowns.
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