Darvish (13-9) scattered six hits and walked two in the gem, during which he twice stranded runners on third base.
The Japanese right-hander tossed 70 of his 110 pitches for strikes as the American League West-leading Rangers clinched their three-game series.
Mike Adams hurled a scoreless eighth and Joe Nathan completed the shutout with his 27th save, retiring the Rays in order in the ninth.
Kinsler's decisive solo shot came in the fourth inning off James Shields (12-8).
The Tampa Bay right-hander absorbed the tough-luck loss, surrendering only three hits over seven innings while walking two and striking out eight.
The Rays lost their fourth straight and fell 1 1/2 games behind American League wild card co-leaders Baltimore and Oakland.