DeRosa stepped up when the Indians, who trailed 4-0 in the sixth, needed him. He hit an 0-1 pitch from Jamey Wright into the seats in right for his fifth career grand slam.
"He threw me two cutters," DeRosa said. "The first one was a little up. I said to myself, 'Don't try to kill the ball, just try to get the guy in from third.'"
DeRosa didn't know if his drive would clear the wall.
"I knew it was deep enough to get over his head," he said. "I was full-go out of the box. I wasn't taking anything for granted."
DeRosa, who leads the team with 42 RBIs, has started games at third base, first base, left field, right field and DH.
"He's done everything we've asked him to," manager Eric Wedge said. "He's had a lot of big hits. He's a baseball player in every sense of the word."
Ben Francisco led off the seventh with an infield hit off Royals reliever Juan Cruz but was forced at second on Jamey Carroll's unsuccessful sacrifice bunt attempt. After Victor Martinez walked, Cruz hit Shin-Soo Choo with a 3-2 pitch to load the bases. Wright came in and gave up DeRosa's big blow.
The Indians lead the Royals by a half-game for fourth place in the AL Central.
Rafael Perez (1-1) was the winning pitcher. He allowed a hit in two-thirds of an inning, striking out two.
The Indians managed only two hits in five innings against Kansas City starter Brian Bannister before erupting in the sixth. The frame started with Carroll's walk and Martinez's single. Choo's fielder's choice scored Carroll, and Martinez was safe at second on first baseman Billy Butler's throwing error. DeRosa's single loaded the bases and ended Bannister's night.
With John Bale pitching, Travis Hafner hit a ground ball to Butler, who threw home, but Miguel Olivo came off the plate for an error as Martinez scored. Kelly Shoppach's sacrifice fly cut the lead to one, and pinch-hitter Jhonny Peralta tied the game with a fielder's choice.
"We had to fight through at-bats to create the opportunity where we were able to get back in the ballgame," Wedge said.
Lee gave up a solo homer to Olivo in the third when the Royals scored twice. Kansas City scored two in the sixth on two singles to the outfield, two infield hits and a throwing error by Shoppach.
"We won the game," Lee said. "That's the thing that matters. In the sixth inning, it seemed like every ball they hit somehow found a way to get them on base. It was a strange inning."