From 1939-1957, Kennedy played for the Chicago White Sox (1939-42, 1946-48, 1955-56, 1957), Cleveland Indians (1948-54), Baltimore Orioles (1954-55), Detroit Tigers (1956) and Brooklyn Dodgers (1957). He batted and threw right-handed. After his playing career, Kennedy managed the Chicago Cubs (1963-65) and Oakland Athletics (1968). His son, former major league catcher Terry Kennedy, was a four-time All-Star and minor league manager.

Kennedy was born in Chicago, Illinois. A line-drive hitter, he was blessed with a strong and accurate throwing arm. On June 22, 1937, the night before the White Sox signed him, Kennedy was working as a 17-year-old popcorn vendor at Comiskey Park during the World Heavyweight Boxing Title between Joe Louis and James J. Braddock. Kennedy debuted a year later, and became the starting third baseman in 1940. After a break of three years to serve in the military during World War II, he returned to play mostly in right field.

In the 1948 midseason Kennedy was sent to Cleveland in the same trade that brought Pat Seerey to Chicago. Kennedy hit .301 the rest of the year and became a member of the last World Championship Indians team. His most productive season came in 1950, when he posted career-highs in batting average (.291), runs (79), hits (157) and doubles (27). The same season, he started two triple plays from the right field, matching Indians left fielder Charlie Jamieson's two triple plays of 1928.

Kennedy was traded to the new Baltimore Orioles in 1954. On July 30, he belted the first grand slam in the franchise history againt Yankees pitcher Allie Reynolds.

In 1955, Kennedy was purchased by the White Sox and sent to Detroit in 1956. Released in April 1957, he signed as a free agent with the White Sox, for his third stint with the club. A month later, he was selected off waivers by the Brooklyn Dodgers, being released at the end of the season.

Following his retirement as a player, Kennedy managed the Cubs from 1963-1965 and posted a 182-198 record in those seasons. In 1968, when the Athletics moved from Kansas City to Oakland, Kennedy was their first manager. Oakland finished sixth in a 10-team league with an 82-80 record, a notable improvement from the 62-99 last-place 1967 Athletics. According to the 1972 book Mustache Gang, authored by Ron Bergman, on the last day of the season Kennedy walked into Charlie Finley's office, expecting an extension. Five minutes later, Kennedy had been fired. After that, he spent four seasons, beginning in 1977, as the Cubs general manager.

In a 16-season career, Kennedy was a .254 hitter with 63 home runs and 514 RBI in 1483 games. As a manager, he posted a 264-278 record in two-plus seasons.

Bob Kennedy died in Mesa, Arizona, at age of 84.