The Royals also have lost eight of 12 against left-handed starters on the road and are 10-13 against them overall.
But Royals manager Ned Yost said he doesn't see a trend.
"It's just one of those things," Yost said. "Not overly concerned. When we're swinging the bats good, our lefties hit lefties pretty well. Moose [Mike Moustakas] hits lefties well, [Alex] Gordon and Hoz [Eric Hosmer] do. I'm just not overly concerned."
Montgomery, a former Royals prospect, struck out 10 while tossing a four-hitter. After the second inning, the Royals simply were overmatched.
"Everything he throws, he throws for strikes," Royals shortstop Alcides Escobar said. "His changeup ... it was really good. He was really, really good tonight."
Hosmer was Montgomery's roommate at Triple-A Omaha.
"We're friends but you still want to win," Hosmer said. "I knew it was just going to be a matter of time for him. He's that good. But hopefully we see him again in Kansas City and get him then."
Royals left-hander Danny Duffy was a teammate of Montgomery's at Double-A Northwest Arkansas.
"He was throwing 98 back then," Duffy said. "Now he has refined it and everything, I mean everything is low in the zone. He pitched great. We just have to answer."
Duffy starts for the Royals on Wednesday.
Jeffrey Flanagan is a reporter for MLB.com. Follow him on Twitter at @FlannyMLB. This story was not subject to the approval of Major League Baseball or its clubs.