KANSAS CITY, Mo. -- After a lukewarm offensive start -- a .244 batting average in April and a .237 mark in May -- the Boston Red Sox are warming up with the summer temperatures.

The Red Sox are batting a major-league-best .292 in June after stroking 16 hits while thrashing the Kansas City Royals 13-2 Sunday.

"I think the extra-base ability up and down the lineup is starting to come through a little bit more regularly," Red Sox manager John Farrell said. "You look over the month so far, we're swinging the bat with more authority and with more like the capability this team has."

Designated hitter David Ortiz hit a milestone home run, and shortstop Xander Bogaerts stroked three doubles and drove in three runs.

The Red Sox set season highs with their run total as well as with eight doubles and 12 extra-base hits.

Left-hander Wade Miley (7-7) stymied the Royals for six scoreless innings, allowing five hits while striking out two and walking three.

Boston center fielder Mookie Betts, third baseman Brock Holt, second baseman Dustin Pedroia and Bogaerts each had three hits. Needing a single to complete the cycle, Betts flied out to center in the ninth.

"I'm not mad about it. It's just what I remember the most," Betts said. "Obviously, having a cycle would be great, but if I hit a double, it would be a double. You just have to play the game.

"I wasn't thinking it, but some of the guys in the dugout were saying, 'When you go back up, (you just need) a single.' A couple of fans were saying, 'Single, single!' But at the end of the day, I was just trying to do the same thing I'd done all day."

The Royals avoided a shutout when center fielder Lorenzo Cain's two-double in the ninth off Koji Uehara scored second baseman Omar Infante and catcher Drew Butera.

Ortiz, Betts and left fielder Hanley Ramirez homered as the Red Sox jumped out to a 7-0 advantage after five innings.

"It's hard to pitch around those guys," Royals manager Ned Yost said. "They are pretty stacked all the way up and down their lineup."

Ortiz led off the fourth with his 476th career home run, passing Hall of Famers Stan Musial and Willie Stargell for 28th on the all-time list.

"It's great," Ortiz said. "I've been playing this game for a long time. As a kid, you dream of watching these guys playing, but you don't dream that far, of having your name right next to them with the numbers. If you have a pretty good career and you work hard, you can get there like they did."

Ramirez, who is among the American League leaders with 15 home runs, turned on a Chris Young 85 mph fastball, propelling it above the Royals' left field bullpen to begin the second inning. The homer provided the first run of the game.

Betts, who has a nine-game hitting streak, homered in the fifth with catcher Sandy Leon aboard.

After Betts' home run, Pedroia doubled, and Young walked Ortiz and Ramirez to load the bases with two outs. Bogaerts cleared them with a double to finish Young's afternoon.

Young, who entered the game with a 1.98 ERA, yielded seven runs on seven hits, all for extra bases, in 4 2/3 innings before being replaced by Jason Frasor.

"In the fifth, I lost it," Young said. "They're a hot-hitting club. That's a good offensive team. I fell behind too many hitters in the fifth. Until then, two solo home runs and the chances are they don't beat you, but in the fifth, the wheels fell off."

NOTES: Red Sox 3B Pablo Sandoval (sprained left ankle) and C Blake Swihart (sprained left foot) were not in the lineup Sunday. Brock Holt started at third, while Sandy Leon caught. 1B Mike Napoli was the emergency catcher. ... Royals C Salvador Perez, who had been behind the plate for 519 of the 576 innings, sat out Sunday, a day game after a night game. ... The Red Sox claimed C Erik Kratz off waivers. The Royals designated Kratz for assignment June 11. ... The Royals, who are 17-14 on the road, play their next nine games on the road, beginning Monday at Seattle, facing RHP Felix Hernandez in the opener. ... The Red Sox return to Fenway Park on Tuesday for a mini-homestand, three games with the Baltimore Orioles.