His fourth failed save opportunity this season came when Chicago's Alexei Ramirez hit a solo homer and Olando Cabrera added a game-winning RBI single in the bottom of the 10th inning.
"We haven't played good baseball, top to bottom, all year," Borowski said after Wednesday's loss. "You're just waiting for it to happen."
His record was 1-3 with a 7.56 ERA and six saves in 10 chances.
Borowski had been the Indians' closer since the beginning of the 2007 season and saved an American League-high 45 games last season.
In the 2007 season, the 37-year-old right-hander went 4-5 with an ERA of 5.07 in 65 2/3 innings but his 45 saves helped the Indians claim the American League's Central Division title.
Cleveland recalled pitchers Jensen Lewis and Brian Slocum from its Class AAA minor league affiliate in Buffalo.