The makeup date was set for Monday, Aug. 25, an open date for both clubs. The Yankees will have to stop through Kansas City on their way from New York to Detroit.
The Royals won the abbreviated series, two games to one, and they've won seven of their last 11 games.
Just after the postponement, the Royals called up left-handed reliever Donnie Joseph from Omaha and sent right-handed reliever Wilking Rodriguez to the Triple-A club.
"Lot of lefties in Cleveland," manager Ned Yost said. "Helps us match up."
Joseph was called up earlier this season, on April 7, and was returned to Omaha two days later without pitching in a game. In 2013, he pitched in six games for the Royals, giving up four hits and four walks in 5 2/3 innings but was not charged with a run.
The Indians open a two-game series at Kauffman Stadium on Tuesday night.
"Their lefties are really good left-handed hitters, and the ability to match up one extra arm out of the bullpen to come in and get one big out or two big outs could benefit us," Yost said.
Yost said that left-hander Bruce Chen, still on an injury rehabilitation assignment with Omaha, was under consideration to be reinstated. He's due for at least one more rehab outing to test his recovery from a bulging disk in his back.
Chen was knocked around in a start against Oklahoma City on Sunday, giving up 10 hits and four runs in four innings. He threw 56 pitches, 40 for strikes, in a 4-3 loss.
"He feels good, he got through it all right -- health-wise," Yost said. "Just so you're healthy, you're feeling good," Yost said. "That's what I'm looking at right now."
Rodriguez, called up from Omaha on June 2, appeared in two games and worked two scoreless innings, giving up one hit and one walk.
"We gave him a look, he was throwing the ball well down in Triple-A. We'll need him again," Yost said.
The Royals simply moved their pitching rotation back a day, meaning that left-hander Jason Vargas, who was due to start against the Yankees, instead will go against the Indians on Tuesday night. Right-hander Yordano Ventura will start on Wednesday night against the Tribe.
After Thursday's open date, the Royals will send Jeremy Guthrie, Danny Duffy and James Shields in that order against the White Sox in Chicago on Friday, Saturday and Sunday.
Because of the off-day, the starters were getting an extra day of rest between starts.
"Now they get an extra-extra day, which is sometimes good, sometimes not," Yost said.
After two straight victories over the Yankees and five wins in the last seven games, the rainout could be seen as a detriment to the Royals' momentum.
"When you're playing well, you really don't like rainouts, but there are benefits to it, too," Yost said. "Now Sal gets his off-day, Omar gets an off-day, Gordy gets an off-day. Play two games and they get another off-day. So there are pros and cons to it."
The references were to catcher Salvador Perez, who'd been scheduled to sit out Monday night's game; second baseman Omar Infante, who's in a 1-for-21 slump; and left fielder Alex Gordon, who's played all 63 games so far.
Shortstop Alcides Escobar, first baseman Eric Hosmer and designated hitter Billy Butler also have played in every game.
"It helps the bullpen, too," Yost said.