The Twins assigned Oliva to the class-D Wytheville Twins in the Appalachian League, where he played in 64 games and led the league with a .410 batting average, but had a low fielding percentage of .854. After finishing second to Orlando Cepeda in batting average in the Puerto Rico leagues in winter ball, Oliva was sent to single-A Charlotte in the South Atlantic League, where he played 127 games and hit .350 with 17 home runs and 93 RBIs. He was called up to the major leagues with nine games left and debuted for the Twins on September 9, 1962, hitting a searing .444 in 12 plate appearances.

In 1963, he was invited to spring training with the Twins and management hoped that the lefty Oliva would counterbalance their right-handed sluggers Bob Allison and Harmon Killebrew. While there, he became friends with teammate, and fellow Cuban, shortstop Zoilo Versalles, who quickly became convinced that Oliva was “the new Ty Cobb”, citing their similarities in hitting ability, speed, and arm strength. However, Oliva failed to make the Twins’ major league team and was assigned to the Dallas-Fort Worth Rangers, the club’s Class AAA affiliate in the Pacific Coast League. Disappointed, Oliva started the season slow, compiling a .235 average in his first two months. He recovered, however, and finished the minor league season with a .304 batting average with 23 home runs and 74 RBI. This earned him a call up for the final few games of the 1963 major league season. Once again he responded with a sizzling bat, hitting .429 in just 7 at-bats.

In 1965, Oliva won a second straight AL batting title with a .321 average, his back-to-back wins once again a baseball first. His performances were all the more noteworthy for falling right at the onset of baseball’s “second deadball era”, with only two other AL hitters reaching the .300 mark that season, (Carl Yastrzemski (.312) and Vic Davalillo (.301)). Oliva added 16 home runs, 98 runs batted in, and 107 runs. He led the AL in hits (185), runs created (108), sacrifice flies (10), and batting average (.321), good for a second-place finish in MVP voting to teammate and Twins sparkplug Versalles. That season, Oliva became an All-Star for the first time. Oliva got into the All-Star game in right field in the eighth inning, replacing game starter Rocky Colavito, and had a double in the ninth inning after pinch hitting in the seventh inning and grounding out.

Oliva was a coach for the Twins after he retired as a player. As such, he holds the distinction of being the only on-field team member to appear with all three Minnesota Twins’ World Series teams: star outfielder in 1965, hitting coach in 1987 and bench coach in 1991.

Through the end of July in 1966, Oliva was leading the league with a .328 average. However, between August 21 and August 28 and September 13 and September 21, he slumped badly, collecting three hits in 29 at bats and four hits in 34 at bats respectively, which cost him a chance at his third straight batting title. Oliva finished with a .307 batting average and was the runner-up to Triple Crown winner and AL MVP Frank Robinson, who hit .316. For the third year in a row, Oliva led the AL in hits (191). Additionally, he won his only Gold Glove award, and finished sixth in MVP voting. One of the season’s highlights came on June 9, 1966, in the seventh inning of a game against the Kansas City Athletics, where Oliva joined Harmon Killebrew, Don Mincher, Rich Rollins and Versalles to hit five home runs in a single inning. These five home runs, hit off starter Catfish Hunter (three) and reliever Paul Lindblad (two), still stand as an MLB record for the most home runs in a single inning.

After a somewhat off 1967, where he led the AL with 37 doubles and finished in the top 10 in batting average, slugging percentage, hits, total bases, RBIs, runs created, extra base hits, and intentional walks, Oliva’s rebound in 1968 was cut short by injury. Missing the last 34 games, he once again hit .289, but so depressed were batting averages it was good for third in the AL by a single point (and only batting champ Yastrzemski topping .300 by a single point as well). He returned to form in 1969 to again place third in the AL with a .309 batting average, with 24 homers, 101 RBIs, and league leads in hits (197) and doubles (39). He finished third in batting average in the American League in 1970 at .325, with 23 home runs and 107 RBIs. He also led the AL in hits (204) for the fifth time, in doubles (36) for the fourth time, and finished second in MVP voting for the second time, this time to Baltimore’s Boog Powell.

Oliva started dating Gordette (DuBois) in the mid-60s. They were married in Hitchcock, South Dakota in 1968 and settled in Bloomington, Minnesota. He currently lives in a house he bought in 1972 and all of his four children live within 10 miles (16 km) of their parents. As of 2011, Oliva also had four grandchildren.

In 1971, Oliva won his third AL batting title with a .337 average and led the league in slugging percentage (.546). These feats at the end of a skein of eight straight All-Star appearances that began his rookie season marked the high point of his career, as severe knee, leg, and shoulder injuries hampered his remaining playing days. His roommate Rod Carew often heard Oliva “moaning and groaning” and getting up to obtain ice for his sore knees during the night. He missed all but ten games of the 1972 season, which required season-ending surgery. Due to injuries and a 1973 American League rule change establishing the position, he became the Twins’ designated hitter that spring and remained in that role his final four seasons.

In a 1976 Esquire magazine article, sportswriter Harry Stein published an “All Time All-Star Argument Starter”, consisting of five ethnic baseball teams. Oliva was the right fielder on Stein’s Latin team. In 1981, Lawrence Ritter and Donald Honig included him in their book The 100 Greatest Baseball Players of All Time. They explained what they called “the Smoky Joe Wood Syndrome”, in which a player of truly exceptional talent but whose career was curtailed by injury, in spite of not having had career statistics that would quantitatively rank him with the all-time greats, should still be included on their list of the 100 greatest players.

The Twins retired Oliva’s uniform number 6 on July 14, 1991. In 2000, Oliva was one of six members of the franchise voted and inducted into the initial class of the Minnesota Twins Hall of Fame. Also inducted in 2000 were teammates Rod Carew and Harmon Killebrew, along with Kirby Puckett, Kent Hrbek and long-time owner Calvin Griffith. On April 8, 2011, the Twins unveiled a statue of Oliva at Target Field coinciding with the team’s 2011 home opener.

Writer Bill James, utilizing the Keltner list, determined that Oliva was a “viable Hall of Fame candidate”, but ultimately did not endorse him as a Hall of Famer. Several contemporaries have endorsed Oliva’s enshrinement in the Hall of Fame, including Tony Pérez, who mentioned in his 2000 induction speech that he hoped Oliva would soon be in the Hall of Fame.

Oliva was considered for the National Baseball Hall of Fame by its Veterans Committee’s election in 2000, but was unsuccessful. He was considered again by the Hall of Fame’s Golden Era Committee election in 2011, but was short by four of the required 12 votes. In 2014, the Golden Era Committee considered him for the second time, but both he and Dick Allen were one vote short of election; Ron Santo was elected in 2011 and none were elected in 2014.