After being a two-sport star in college, Jensen was the first person to play in the Rose Bowl, the World Series, and the MLB All-Star Game.

Cal ended the regular season at 10–0 under coach Pappy Waldorf, winning a share of its first Pacific Coast Conference title in ten years, and Jensen placed fourth in the Heisman Trophy voting, with Doak Walker taking the award.

In the 1949 Rose Bowl, Jensen scored a touchdown in the first quarter to tie the game 7–7, but fourth-ranked Cal was upset 20–14 by seventh-ranked Northwestern.

In 1949, Jensen, who batted and threw right-handed, left college after his junior year and signed with the Oakland Oaks in the Pacific Coast League.

His contract – along with Billy Martin's – was sold to the New York Yankees in 1950 with the intention of him being a backup for future Hall of Fame centerfielder Joe DiMaggio.

He finished the season with a .286 batting average and 80 RBIs, leading the league with 17 assists and placing third in the AL with 18 steals, a total he duplicated in 1953.

In June of that same year, he set a Red Sox club record for most home runs in a single month (since tied by David Ortiz) with 14, and in July made his final All-Star team.

Smack in his productive prime at 32, Jensen announced his retirement from baseball in January 1960, primarily because of an intense fear of flying, but also the result of the long separations from his family.

Indeed, upon his trade to the Red Sox in 1953 he considered not reporting to the team in order to return to his family in California, but general manager Joe Cronin increased his salary by $1,000 and he agreed to play.

As teams had increasingly turned to air travel in the 1950s, he had unsuccessfully sought to combat his aversion to flying, aided considerably by Red Sox owner Tom Yawkey, who arranged for therapy treatments.

Once major-league baseball expanded to the West Coast in 1958, and with further expansion and constant air travel foreseen, Jensen's difficulties became virtually insurmountable.