The Royals battled back from a five-run deficit and capitalized on a costly error by shortstop Yunel Escobar that led to four unearned runs.
Tampa Bay held a 6-4 lead heading into the Kansas City sixth, but the Royals loaded the bases with one out against Jake McGee (0-2) and Billy Butler then drove in a run with a ground out to the pitcher.
Eric Hosmer followed with a grounder to Escobar at shortstop, but he misplayed the ball for an error on what should have been the third out of the inning.
One run scored on that play, Lorenzo Cain followed with an RBI single and Francoeur drove in two more runs with a base hit.
Bruce Chen (2-0) picked up the win with two innings of scoreless relief. He struck out three and gave up two hits. Greg Holland worked a perfect ninth for his seventh save.