Cron extended his on-base streak to 22 games when he was hit by a pitch from Jason Hammel (0-5) in a three-run first that included Brad Miller's RBI groundout and Adeiny Hechavarria's two-run single.
Cron homered leading off the third, then singled in the fifth and scored on Wilson Ramos' double-play grounder for a 5-1 lead.
Tampa Bay had losing six of seven coming in but swept the Royals for the first time since four games from Aug. 8-11, 2011. The Rays had not swept a series in Kansas City since three games from July 17-19, 2009.
Kansas City has lost five straight and seven of eight, dropping to 13-30 for its worst 43-game start since opening with a 10-33 record in 2006. The Royals are 6-16 at Kauffman Stadium.
Matt Andriese (1-1) allowed three hits in three scoreless innings. Alex Colome allowed Abraham Almonte's leadoff single in the ninth, then retired three straight batters for his third save in three days and ninth in 11 opportunities.
Staked to a 4-0 lead, Rays starter Jake Faria allowed three runs, four hits and four walks in 4 1/3 innings. He gave up sacrifice flies to Alcides Escobar in the fourth and Mike Moustakas in the fifth, an inning that also included Jorge Soler's RBI double.
Hammel gave up five runs and 10 hits in six innings, raising his ERA to 6.28. He is 0-8 with a 7.28 ERA in 13 starts since winning at Detroit on Sept. 6.
LONG TIME COMING
LHP Jonny Venters went 2,096 days between victories. He picked up the win Tuesday in relief, his first since Aug. 17, 2012, while with Atlanta. There were 1,058 other pitchers who got wins between his two victories. ''Every time I hear numbers like that, it's funny to sit back and think about it,'' said Venters, who has had three Tommy John surgeries and missed nearly four seasons.
Tampa Bay recalled INF Christian Arroyo from Triple-A Durham, where he hit .200 in 17 games.
Rays: OF Carlos Gomez (right groin strain) was placed on the 10-day DL. ... 3B Matt Duffy (right hamstring tightness) did not play. ''I think we're probably going to play it a little safer,'' Duffy said. ''If I need one extra day, the first day in Anaheim, I don't think that's out of the question, to take two days off and go from there.''
Royals: INF Cheslor Cuthbert (back spasms) was not in the lineup for the second straight day. ... RHP Nathan Karns (right elbow inflammation) has resumed a throwing program after being shut down for several days.
Rays: RHP Chris Archer (2-3) is to start Thursday at the Los Angeles Angels. He gave up three homers and six runs in a loss Saturday at Baltimore and is 5-1 with a 2.55 ERA in seven starts against the Angels.
Royals: RHP Jakob Junis (4-3) is to start Friday in the opener of a three-game series against the visiting New York Yankees.
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