A native of Kenosha, Wisconsin; Jensen was a pitcher for the Minneapolis Millers before an injury ended his baseball career. He then took a job as a sportswriter at the Minneapolis Tribune (now part of the Minneapolis Star Tribune), then moved into broadcasting--first at WLIP-AM in Kenosha, then at WCCO-AM in Minneapolis, WMBD-TV in Peoria, Illinois; and WBZ-AM-FM-TV in Boston.
In November 1963, he was the "pool" reporter in Hyannis Port at the Kennedy compound in the aftermath of John F. Kennedy's assassination. His reporting was noticed by WCBS, who hired him in 1964. He soon became weekend anchorman and backup weekday anchor behind Robert Trout, who was doing double duty at the station and at CBS News. Jensen didn't take too long to make an impact, winning notoriety for his coverage of Robert F. Kennedy's Senate campaign soon after he arrived in New York. When Trout left for a network assignment in Europe, Jensen succeeded him as WCBS' main anchor. He was the face of the WCBS newsroom for the next four decades.
Jensen was known in New York for his booming, gravelly voice and deliberate demeanor, and was often thought of as a local version of Walter Cronkite. WCBS had gone back and forth with WNBC-TV for first place, but under Jensen became the dominant station in New York, a lead it kept for most of the time until the mid-1980s. He was also known for asking perceptive questions, even of his colleagues at the news desk. WCBS' reporters had to know their stories very well if their stories aired when Jensen was behind the anchor desk. They risked embarrassing themselves on-air if Jensen asked them a question that they couldn't answer. Over the years, his partners at the anchor desk – some of them New York broadcast legends in their own right – included Ralph Penza, Rolland Smith, Carol Martin, Michele Marsh and finally Dana Tyler. He was reportedly the model for Jim Dial, Murphy Brown's co-anchor.
In 1994, WCBS demoted Jensen to host of its Sunday morning public-affairs show. At that time, he had been WCBS' lead anchor for 29 years--longer than anyone in New York television history. The station's ratings had tailed off considerably, and management wanted new blood at the anchor desk. However, the decision and the manner in which it was handled caused a firestorm of criticism, which only increased when Jensen was forced to retire in 1995 shortly after Westinghouse announced it was buying CBS. WCBS' ratings plummeted even further, and by the end of 1995 it had crashed into last place and stayed there for more than a decade before recovering in the mid-2000s.