Tampa Bay's slugging third baseman Evan Longoria went 3-for-4 with a single, a double and a two-run home run in the third inning to support reigning Cy Young winner David Price, who hurled a complete game seven-hitter for the win.
Victory in the tiebreaker, which was needed after the two teams finished the regular season with 91-71 records, gave the Rays the American League's second wild card berth.
They will face the Cleveland Indians in a one-off game on Wednesday with the winner to play a best-of-five Division Series against the Boston Red Sox.
There was also a measure of revenge for Tampa Bay, who had been eliminated by Texas in the Division Series in their last two playoff appearances in 2010 and 2011.
Price improved to 10-8 in the game that counted as the 163rd game of the regular season, while the Rangers' 22-year-old rookie left-hander Martin Perez of Venezuela dropped to 10-8 after giving up three runs in five and one-third innings.