A late tiebreaking home run from Yoenis Cespedes helps, too.
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Cespedes hit a leadoff homer in the eighth inning and the A's beat the Royals 4-3 to move back over .500.
"I would rather see us win by 10 runs ... but it makes for a little bit more excitement, certainly at the end," Oakland manager Bob Melvin said. "What it does for me more than anything is it shows some tenacity that when you get down, you don't worry about it."
Cespedes hit his eighth homer, connecting against Kelvin Herrera (2-4). Cespedes also singled and scored the A's first run in the second.
"I kind of hesitated, pulled up and it took one of those sharp hops and just got by me on the top of my glove," Lough said. "If I would have went for it I would have caught it in the air, for sure."
Still, Oakland's first sweep of the Royals since 2008 didn't come easily.
The A's squeaked out a pair of 2-1 wins on Friday and Saturday, then had to sweat out another tight finish in the series finale.
Cook, pitching because closer Grant Balfour was unavailable, allowed two-out singles to Billy Butler and Eric Hosmer in the ninth. Mike Moustakas then worked a full count off Cook before grounding out to end the game.
"Any time you can get three wins and sweep a team that's playing pretty decent baseball, it doesn't really matter how you do it," Norris said. "It's definitely a plus for us."
Alex Gordon matched his career high of four hits for Kansas City. The Royals have lost 10 of 13 and fallen back to .500.
Gordon hit an RBI single in the third, scored in the fifth and doubled in the seventh. Since moving to the third spot in the lineup on May 9, Gordon is batting .450 (18 for 40).
The Royals, who scored just two runs and stranded 11 baserunners in the first two games of this series, went up 2-0 when Chris Getz singled in Hosmer in the fourth. Hosmer added an RBI double in the fifth.
A's starter A.J. Griffin had command problems most of the day. He walked three and struck out five but failed in his bid to become the second pitcher in Oakland history to reach 12 wins in his first 24 starts.
- Each of the A's last 12 home runs have been solo shots, the second-longest such streak in Oakland history.
- Mendoza struck out four and didn't walk a batter for a second straight start.
- Gordon had 60 multihit games in 2012 but already has 20 in the Royals first 40 games this season. He has nine career four-hit games.
- Injured Oakland OF Josh Reddick continues to make slow progress from surgery on his right wrist. He was out Sunday after taking swings off a tee the previous day and is eligible to come off the disabled list Wednesday, though manager Bob Melvin thinks Reddick will need more time to heal.
- Kansas City RHP Jeremy Guthrie (5-1) pitches the series opener in Houston and is seeking to rebound from his first loss of the season.
- A's RHP Bartolo Colon (3-2), who takes the hill against Texas on Monday, is winless in his last four starts.