More importantly, they think they dodged an injury that could have significantly hampered their chances in October.
After allowing one run and two hits in the first six innings, Twins starter Jake Odorizzi caught manager Rocco Baldelli's attention with some odd arm movements as he warmed up for the seventh. Odorizzi's arm wasn't the problem.
''He asked about my shoulder and I said that was fine - I was just getting the blood flow going again - but my hamstring was cramping a little,'' Odorizzi said. ''I could have kept going, but taking me out was probably the smart thing.''
Baldelli agreed and brought in Kyle Gibson.
''Under different circumstances, I would have left Jake out there, but this way we are sure he's going to be fine going forward,'' he said.
Minnesota's magic number to clinch the division is two over Cleveland. The Indians beat the White Sox 11-0 later Tuesday.
Odorizzi (15-7) hadn't allowed a hit since Harold Castro's bunt single in the first and left having struck out six and allowed one run.
''I felt great,'' he said. ''I just need to hydrate a little more.''
The Twins improved to 97-60, their most victories since going 98-64 in 1970. With three wins in their final five games, they can become the record fourth club with 100 wins this season.
Taylor Rogers pitched a perfect ninth for his 29th save. It was his first appearance since allowing three runs in a loss to Kansas City on Saturday.
Detroit (46-110) reached 110 losses for the second time, 16 years after setting the AL record of 119. With one more loss or a victory by Baltimore, the club would be assured the No. 1 pick in next year's amateur draft.
''The Twins score millions of runs and we held them to four,'' Tigers manager Ron Gardenhire said. ''We had chances, too. We definitely had chances.''
The Tigers need to win one of their final two home games to avoid tying the 1939 St. Louis Browns' record of 59 home losses.
The delay that started the confusion over Odorizzi's health was caused by Minnesota's four-run seventh inning.
Tigers starter Spencer Turnbull started the game with six scoreless innings, but couldn't get an out in the seventh. Jake Cave led off with a single and LaMonte Wade Jr. went the other way to beat the shift for a double.
''It is tough, because I didn't leave the guys with a good situation,'' Turnbull said. ''I think I did pretty well, but it is hard to be excited when you leave with two guys on and no one out in a 1-0 game.''
Astudillo hit David McKay's third pitch up the middle for a two-run single, ending Turnbull's chance at a win. Rosario added a two-run double off Nick Ramirez to make it 4-1.
''The guys in the lower part of the order were getting good at-bats all night, but we couldn't string anything together until they got us going in the seventh,'' Baldelli said.
Turnbull (3-16) allowed six hits and two walks in six-plus innings. Turnbull has an 0-12 career record at home - the most losses by any major league pitcher without a home win.
The Tigers scored in the first. Dawel Lugo doubled, took third on Castro's bunt and scored on Jeimer Candelario's sacrifice fly.
The Tigers put runners on the corners in the seventh, but Gibson struck out Jake Rogers to end the inning. Lugo made it 4-2 with an RBI double in the eighth.
Twins: 1B Marwin Gonzalez was a late scratch due to tightness in his right oblique muscle. He was replaced by Astudillo and is listed as day-to-day.
Tigers: DH Miguel Cabrera will have his right knee examined by several specialists after the season in the hopes of finding a solution to the chronic pain that turned him into a full-time DH this season. Cabrera has missed the last three games, but hopes to play Wednesday.
Gibson's two innings of relief might have been a preview of his role in October. Gibson isn't at full strength while dealing with ulcerative colitis.
''We wanted to see what he could do throwing all out for a couple innings instead of having to grind through six or seven,'' Baldelli said. ''We're planning to give him one more outing like that before the end of the regular season.''
The teams play the second of a three-game series Wednesday evening, with Detroit's Daniel Norris (3-13, 4.58) scheduled to open against Twins rookie Randy Dobnak (1-1, 2.01).
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