Cleveland was one strike away from ending its skid when closer Emmanuel Clase threw a wild pitch, allowing pinch-runner Zach McKinstry to score the tying run with two outs in the ninth.
The Guardians' double-digit streak is the 11th in team history and the first since Cleveland dropped 11 in a row in 2012. The franchise record is 12 in 1931.
Cleveland’s Steven Kwan had three hits, including an RBI double in the eighth.
Báez singled leading off the 10th with automatic runner Parker Meadows aboard and Sweeney followed with a three-run drive down the left-field line off Cade Smith (2-3) for a 4-1 lead.
Skubal — the reigning AL Cy Young winner and possible starter in next week's All-Star Game — allowed three hits in seven innings. It was his sixth double-digit strikeout game of the season and 13th of his career.
Chase Lee (4-0) pitched 1 2/3 scoreless innings.
Cleveland's Gavin Williams allowed one hit in six innings and struck out eight.
Key moment
The Tigers padded their lead in the 10th on Matt Vierling's RBI double and Greene's two-run homer, his 22nd.
Key stat
Greene is the 15th Detroit player to hit at least 22 homers before the All-Star break. The last was J.D. Martinez, who had 25 in 2015.
Up next
Tigers: Have not named a starter for Monday's series opener against visiting Tampa Bay.
Guardians: Open a six-game trip at Houston on Monday. RHP Tanner Bibee (4-9, 4.20 ERA) goes up against Astros RHP Hunter Brown (9-3, 1.82).
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