Roberts collected a two-out single against David Robertson (1-5) in a 3-3 tie game, stole second and came home with the decisive run when Gimenez grounded a ball into right field.
The Tampa Bay catcher finished with two RBI while B.J. Upton socked a solo homer as the Rays took the opener of a three-game series.
James Shields (13-8) limited the Yankees to three runs on five hits over eighth innings to earn the victory. He walked three and struck out five.
The third-place Rays moved to within 2 1/2 games of the American League East-leading Yankees with the triumph.
New York starter CC Sabathia yielded three runs on eight hits over seven full innings.
Russell Martin went 2-for-3 with an RBI to cap a three-run Yankees fourth inning.