The Athletics fell to Detroit on Thursday after taking a three-run lead into the ninth inning and they let another lead slip away in this game when the Rays scored twice in the top of the eighth to forge a 3-3 tie.
Price was replaced by Joel Peralta, who retired Josh Donaldson on a grounder to shortstop. Crisp went to second on that play and then scored when Lowrie doubled to right.
The rest of the Oakland scoring came from Kurt Suzuki, who hit a three-run homer in the fifth.
Ryan Cook (6-3), who gave up two hits while Tampa Bay was tying the game in the eighth, emerged as the winning pitcher with his one inning of relief work. Grant Balfour survived a one-out walk in the ninth to collect his 34th save.
Price (8-6) was charged with all the Oakland runs, one of which was unearned, in seven-plus innings of work.
Oakland moved within two games of Texas for the lead in the American League West.