DeRosa's slam off reliever Jamey Wright led Cleveland to an 8-4 victory over the Kansas City Royals in a matchup of the two worst teams in the division. The Indians scored four runs in each of the sixth and seventh innings to move a half-game ahead of the Royals, who have lost 10 of their last 11 and 22 of their last 28.
Cleveland had been in last place since April 8.
"We've battled hard," DeRosa said. "But at the same time, we feel like we're capable of way more than being one game above the cellar."
Rafael Perez (1-1) pitched two-thirds of an inning for the win.
The Indians loaded the bases off Juan Cruz (3-2) in the seventh with a fielder's choice, walk and a hit batsman. Cruz, who began the year as Kansas City's top setup man, has allowed eight earned runs over his last three innings, spanning four appearances.
Wright relieved Cruz, and DeRosa hit his second pitch into the seats in right field for his 10th home run of the season. It was the fifth grand slam of DeRosa's career and third for the Indians this season.
"I just wanted to get the guy in from third," DeRosa said. "Sometimes you try so hard to get everybody in, you end up failing. Whenever you come up in those situations in a tie game, you want to be sure to get one in."
He did much better than that. Wright threw two cutters to DeRosa. The first blew past him for a strike. The second landed in the right field seats.
"The first pitch was a lot worse than that one," Wright said. "It was away, it was just up a little bit. He got it up in the air and it kept going."
DeRosa, acquired from the Chicago Cubs during the winter, has played first base, third base, right field and left field for the Indians. He now leads the team with 42 RBIs.
"He's had a lot of big hits this year," Indians manager Eric Wedge said. "He's done everything we've asked him to do. He's played a lot of different positions and filled different spots in the order. He's a baseball player in every sense of the word."
A pair of Kansas City errors in the sixth allowed Cleveland to tie the score at 4. With runners at first and second, first baseman Billy Butler's throwing error while trying to start a double play loaded the bases and ended starter Brian Bannister's night.
Travis Hafner bounced to first off reliever John Bale, but catcher Miguel Olivo pulled his foot off the plate too quickly, allowing Victor Martinez to score. Instead of being out of the inning, the Indians later tied the game on a sacrifice fly and fielder's choice.
"That's as frustrating as it gets," Royals manager Trey Hillman said. "Defensive letdown. We have to execute. It's that simple."
Bannister was continuing his mastery of the Indians until his defense failed him. Bannister, 3-0 in five starts at Progressive Field and 4-1 in his career against the Indians, had allowed one baserunner to reach second base through the first five innings, but he failed to register an out in the sixth. Only one of the four runs charged to him was earned, giving him a career ERA of 2.34 against Cleveland.
Indians starter Cliff Lee allowed four runs and 11 hits in six innings. It is the most runs he has allowed since he surrendered four against Toronto on April 11 and the most hits he has given up since allowing 12 to Detroit on May 3.
"We had a chance to beat a really good pitcher tonight," Hillman said. "And we let it slip away."
Miguel Olivo homered in the third and Willie Bloomquist stole three bases for Kansas City. The eighth and ninth hitters in the Royals' lineup went a combined 5 for 7 with a home run, three runs scored and four stolen bases. Kansas City was 5 for 5 stealing on the night.
The Indians selected North Carolina RHP Alex White 15th overall in the MLB draft ... SS Jhonny Peralta was not in the starting lineup because of his 2-for-17 career mark against Bannister, although he entered the game in the sixth ... The Royals selected RHP Aaron Crow with the ninth overall pick. Crow was drafted in the first round by the Nationals last year, but did not sign ... OF Coco Crisp returned to the lineup after missing two games with a sore right shoulder ... The Royals requested unconditional release waivers on LHP Horacio Ramirez.