Burrows was born in Toronto, Ontario. Burrows learned to skate at the Pine Point Outdoor Arena in nearby Etobicoke. He won his first Provincial Championship with the Bantam Newmarket Optimists in 1963. As a teenager Burrows was selected the Most Valuable Defenseman and a First Team All-Star with the Dixie Beehives of the Metro Jr. B League. Originally the property of the Chicago Blackhawks organization, Burrows played his junior hockey with the Ontario Hockey Association's St. Catharines Black Hawks before turning pro with the Central Hockey League's Dallas Black Hawks in 1969. The defenseman would ultimately never play a game in Chicago, being claimed by the Pittsburgh Penguins in the NHL's intra-league draft in June 1971. Burrows' hero while growing up was Tim Horton. Burrows had the opportunity to play as the defence partner of his idol for one year, when Horton joined the Penguins for the 1971–72 season. Burrows credits Horton and Red Kelly, his coach that season, for helping him become successful at the NHL level. Burrows was selected as the Penguins "rookie of the year" for 1971-72. In his third season, he was chosen as the Penguins' team MVP. In 1976, Burrows was selected to represent Canada in the Canada Cup Tournament. Chosen as a defensive replacement for future Hockey Hall of Fame member Bobby Orr, Burrows did not play in the tournament as Orr earned MVP honours. A two-time all-star as a Penguin, Burrows was traded to Toronto for Randy Carlyle and George Ferguson in the spring of 1978. After two and a half seasons playing in his hometown, the defenceman was traded back to Pittsburgh along with Paul Gardner in exchange for Kim Davis and Paul Marshall early in the 1980-81 season. Burrows is a member of the Penguins "all-time team", a list of the 15 greatest players in Penguins history as chosen by Trib Total Media. He is third all-time in games played among Penguins defensemen.
On June 8, 1971 the Chicago Blackhawks lost Dave Burrows to the Pittsburgh Penguins in an intra-league draft.
On September 8, 1971 the Pittsburgh Penguins resigned Dave Burrows.
On June 13, 1978 the Pittsburgh Penguins traded Dave Burrows and a 1978 sixth-round pick to the Toronto Maple Leafs in exchange for Randy Carlyle and George Ferguson.
On November 18, 1980 the Toronto Maple Leafs traded Dave Burrows and Paul Gardner to the Pittsburgh Penguins in exchange for Paul Marshall and Kim Davis.
