Mincher was a backup for Vic Power and Bob Allison at the beginning of his career, and gradually won more playing time. He hit 23 HR in just 287 at-bats in 1964 and earned the lead position in a first-base platoon for the pennant-winning 1965 team. He hit .251 with 22 HR and 66 RBI in 346 at-bats. In Game One of the World Series, he homered in his first WS at-bat, off Don Drysdale in the second inning, to give the Twins a lead they never surrendered. He tied a WS record for first basemen with four assists in Game Two as the Twins won again. However, he hit only .130 for the Series.
On June 9, 1966, Mincher contributed one of the Twins’ AL-record five homers in one inning. But Mincher’s power dropped off in 1966 (14 HR in 431 at-bats), and Minnesota traded him to the Angels after the season. He rebounded in 1967 with 25 HR (tied for fifth in the AL) and 81 runs, but on April 11, 1968 Mincher was beaned by Sam McDowell and suffered dizzy spells all season. Hitting only 13 HR, he was left unprotected in the expansion draft. Picked by the Pilots, he came back once again with 25 HR and a career-high 78 RBI, and the A’s traded four players to acquire him and a throw-in for 1970. He hit a personal-best 27 HR that year, but led AL first basemen in errors.
Mincher’s power dropped in the next two seasons as he was dealt to the Senators, who then moved to Texas (he is the only person to play for both Senators teams and the Twins and Rangers). The A’s reacquired him for their 1972 pennant drive, and although he hit only .148, there was one moment of glory left for him. In Game Four of the World Series against the Reds, the A’s trailed 2-1 in the ninth inning with one out and two on. Mincher’s pinch-hit single off Reds bullpen ace Clay Carroll tied the score as the A’s went on to win the game and the Series. It was Mincher’s final major league at-bat.