Francoeur's eighth-inning blast capped the game's scoring while Betancourt's clout in the third completed a four-run uprising that put Kansas City in charge.
Alex Gordon and Mike Moustakas added RBI as the Royals clinched their three-game series by outscoring Tampa Bay 16-2 in the first two contests.
Bruce Chen (7-6) gave up one earned run on five hits through seven innings, striking out five while issuing one walk. The left-hander tossed 72 of his 99 pitches for strikes.
Rays starter Chris Archer (0-2) went 5 2/3 innings, surrendering four runs (three earned) on four hits while striking out seven and walking one.
Brooks Conrad hit a solo homer for Tampa Bay.