The Giants announced the move on Wednesday as they started preparations for the season opener against the Cowboys in Dallas on Sunday.
Brinkley was released by the Cowboys on Monday. He has played in 79 regular-season games with 33 starts for Minnesota (2009-12 and 2014) and Arizona (2013). He missed the 2011 season because of a hip injury.
His career totals include 270 tackles, a sack, five passes defensed and five forced fumbles in the regular season. He also has 32 special teams tackles.
Brinkley was a fifth-round draft choice by the Vikings in 2009.