Kinsler belted a solo homer in the first inning and added a two-run single in the third for a 3-0 Texas lead.
Andrus followed him with a two-run shot and later provided a sacrifice fly as the Rangers notched only their second win in 12 games overall.
Texas and Tampa Bay finished the night holding the two American League wild card spots with identical records (82-68).
Cleveland came into Tuesday's action 1 1/2 games off the wild card pace, while Baltimore bested Boston to pull to within two games of the co-leaders.
Alexi Ogando (7-4) went five innings to get the win, allowing one run on two hits. He walked one and struck out four.
Rays starter Jeremy Hellickson (11-9) was lit up for five runs on four hits, lasting just 2 2/3 innings.
Jose Lobaton homered for the Tampa Bay.