Cody Stavenhagen of The Athletic first reported the deal was done.
The 63-year-old Gardenhire retired Sept. 20 because of health issues.
"It's been wonderful here, but I also know I have to take care of myself," he told reporters. "When you come to the ballpark and you're stressed out all day and your hands are shaking, that's not fun. I've got grandbabies. I've got kids that I need to take care of and my wife."
Gardenhire has had both cancer and diabetes and dealt with stomach issues in the days leading up to his decision.
"This is tough. It's a tough day for me. Didn't expect it, tell you the truth, when I walked in," he added. "But I just know how I've been feeling lately, and I expressed that to Al and elected to just go ahead and step down."
Hinch will take over a Tigers team that is rebuilding after losing 114 games in 2019 before going 23-35 this year. Lloyd McClendon took over for Gardenhire for the remainder of the season as the interim manager and was a candidate to be hired for the full-time gig, though Detroit opted to go with Hinch.
The 46-year-old brings plenty of experience to the position with a background as the manager for the Arizona Diamondbacks (2009-10) and Houston Astros (2015-19). His run in Houston included four trips to the postseason, including a World Series title in 2017 and a trip to the World Series in 2019, though it ended in controversy after he was fired for his involvement in the sign-stealing scandal.
He'll be tasked with taking the Tigers from rebuilding to contending. It's no small job, but with exciting pitching prospects like Casey Mize and Tarik Skubal to build around, among others, the team has reason to be excited about the future.