The Rangers and Rays are tied for the two wild card berths in the American League, but they are only a half game in front of Cleveland. Baltimore is two games back, Kansas City trails by three and New York is 3 1/2 behind.
Texas had lost seven in a row before taking two of the final three games of this four-game series. The Rangers now move on to Kansas City for three and Tampa Bay hosts Baltimore for four with both those series likely going a long way to determining the wild card spots.
Darvish (13-9) has been victimized by a lack of run support this year, but the Rangers piled up 16 hits against the Rays -- four of them home runs.
Andrus drove in two more runs with a single in the fourth. The other Texas home run came from Geovany Soto in the ninth.
Darvish needed 109 pitches to get through just five innings. He gave up five hits and walked six, but held the Rays to two runs in the first.
Texas roughed up Matt Moore (15-4) for five runs on seven hits in just four innings. Moore spent all of August on the disabled list and had not lost a game since June 14.