Sawyer was born in Akron. A graduate of Western Reserve Academy in Hudson, OH, Sawyer received a bachelor of arts degree from the University of Akron in 1968 and a master of arts degree from the same university in 1970.

Sawyer worked as a teacher and then was elected to the Ohio House of Representatives, where he served from 1977 to 1983. In late 1983, he was elected Mayor of Akron.

After only one term as mayor, he was elected to the House from Ohio's 14th Congressional District, based in Akron. In 1984, he bet mustaches with Akron-area Republican Tom Watkins on the outcome of the Mondale / Reagan presidential election; he has been cleanshaven ever since. He was reelected seven times, never facing serious opposition in the heavily Democratic district. However, he was severely criticized for his support of NAFTA.

The redistricting following the 2000 census eliminated Sawyer's district. Most of Akron was placed in the 13th District, represented by fellow Democrat Sherrod Brown. However, Sawyer's home was placed in the Youngstown-based 17th District. The district had been held by Democrat Jim Traficant until his expulsion from the House in 2002. Traficant tried to run for his old seat as an independent. Traficant's former protege Tim Ryan also ran for the seat. The new 17th district was much more heavily pro-labor than Sawyer's old district and Sawyer was seen as being insufficiently pro-labor as a result of his support of free trade. In a surprise upset, Sawyer lost the 2002 Democratic primary to Ryan, who went on to win the seat in the general election.

Sawyer is a 2006 candidate for Ohio's 13th Congressional District. That seat's incumbant, Sherrod Brown, will not be running for re-election and will instead seek the Democratic nomination for U.S. Senate to run against Mike Dewine.

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