Romo had a fourth scoring pass overturned on replay. Instead, Murray wound up with his seventh rushing touchdown of the season on a 1-yard plunge.
Murray finished with 128 yards rushing to pass Brown, who hit the century mark in the first six games of the 1958 season for Cleveland.
The Cowboys (6-1) are off to their best start since they went 13-3 in 2007 and were the top seed in the NFC before losing to New York in their first playoff game.
Eli Manning had three touchdown passes for the Giants (3-4), who have lost to the NFC East's top two teams in consecutive weeks.
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