Bauer was originally teamed up with Dumart and Schmidt while in Providence where their coach Albert Leduc named them the Sauerkraut Line which was eventually shortened to be just the "Kraut Line." Bobby played his minor hockey with the St. Michael's Majors in 1933-34 and the Kitchener Greenshirts in 1934-35.

The Bruins claimed him from Syracuse in the ithe IAHL Inter-League Draft in 1935 and was sent to the Boston Cubs in 1935-36. Never a league leader during those seasons, he nonetheless caught the attention of the Bruins management and he was brought up for the final game of the 1936-37 where he scored his first NHL goal and earned a spot for good with the Bruins.

The Kraut Line was reunited in 1937-38 and quickly became one of the most potent scoring lines in the NHL. So potent a force were they, that in the 1939-40 season they finished 1-2-3 in the scoring race with Schmidt on top with 52 points and Dumart and Bauer tied at 43 points each. That same season Bauer won the first of 3 Lady Byng Trophies. As throughout their career the Kraut Line seemed to do everything together, so they joined the military for World War II. Bauer was the first to go, so from 1941 to 1945 Bauer did not play in the NHL. Again, just as his line mates did, Bobby returned from the war and proceeded to have the best season of his career in 1946-47 with 30 goals and 24 assists for 54 points. That same year he also won his last of 3 Lady Byng Trophies.

Nagging shoulder injuries forced Bauer to retire from the NHL at the end of the 1947 season. But he didn't retire from hockey. He had his amateur status reinstated and played 3 seasons with the Kitchener Dutchmen of the Ontario Hockey League Seniors.

On March 18, 1952, the Boston fans held a "Kraut Night" and to celebrate the evening, Bauer signed a one-game NHL contract and the famous trio was once again reunited. Schmidt scored his 200th career goal that night as Boston downed the Black Hawks 4-0. Late that same season Bauer became the president, manager and head coach of the Kitchener Dutchmen and led them to two OHA Senior Championships and two Allen Cups. That second Allen Cup team was chosen to represent Canada in the 1956 Winter Olympics in Cortina Italy where they won the bronze medal. Following those games Bauer retired, only to be convinced 4 years later to once again lead the Canadians at the 1960 Squaw Valley Olympics where they won the silver medal.

Following the 1960 Olympics Bobby began to share his coaching secrets and National Team vision with his younger brother Father David Bauer who later became a coaching sensation at St. Michael's School for Boys, and would eventually shape the look of the Canadian National Team.