Darvish (9-4), who came off the disabled list with an upper back strain before the contest, showed few effects from the two-week layoff, limiting the Yankees to just two hits over 6 1/3 innings.
The right-hander walked two and struck out four, staking the Rangers to a victory in the opener of a four-game series.
Robbie Ross and Tanner Scheppers combined for 1 2/3 frames of scoreless relief and Joe Nathan wrapped up the victory with his 31st save for Texas, which came into the contest trailing American League West-leading Oakland by three games.
Ivan Nova (4-3) was tagged with the loss, surrendering three runs on seven hits over seven full frames.
Celebrity Families of 2014 [PHOTOS]