Dallas also placed OT Jordan Mills on waivers in a subsequent move to clear a roster spot for their new addition.
Browns actually tried out for the Cowboys last Wednesday, so he was clearly on their radar.
Brown, 28, is a former second-round pick of the New Orleans Saints back in 2010. He spent four years in New Orleans before he signed on with the Giants as an unrestricted free agent in March of last year.
However, Brown lasted less than a year with the Giants, as he was later cut loose in November.
The Jets gave him a shot this summer, but he was among their roster cuts from last week.
In 2014, Brown appeared in three games, but was limited to playing just 62 total snaps, according to Pro Football Focus.