The Indians were shut out for the third straight game, matching a club mark and one shy of the major-league record.
Jack Morris (8-5) won his fifth consecutive start as the Twins surpassed their previous record streak of 12 straight wins in 1980. Morris pitched three-hit ball for seven innings and improved to 29-9 lifetime against Cleveland.
Mack hit his first career grand slam, doubled and singled. He set a career-high for RBI, and is 11-for-29 during the Twins' surge. Kent Hrbek added his sixth home run.
The Indians have only eight hits in their past three games. They went 23 innings without getting a runner past second base until Mark Lewis reached third on a wild pitch in the sixth.
Cleveland has been held to two runs on less in 11 of its last 12 games. The major league mark of four consecutive shutouts in shared by several teams in both leagues.