Kazurinsky was born in Johnstown, Pennsylvania. Kazurinsky started out in St. Louis, Missouri as a copywriter for a department store chain. Kazurinsky was a fairly versatile player on the show, even though he clashed often with then-executive producer Dick Ebersol about the show's creative direction.

• Dr. Jack Badofsky, a doctor appearing on Weekend Update (then called SNL Newsbreak or Saturday Night News) with a list of punny names of diseases.
• Mr. Landlord from Eddie Murphy's "Mr. Robinson's Neighborhood"
• Father Timothy Owens, an Irish priest
• Havnagootiim Vishnuuerheer, a Hindu wiseman who answered audience questions
• Worthington Clotman, SNL's resident network censor who would interrupt sketches and point out any and all objectionable material that needed to be changed. Based on real-life network censor at the time, Bill Clotworthy.
• The husband to a chimpanzee on "I Married A Monkey"

• Mahatma Gandhi (in a movie trailer parody called "Gandhi and the Bandit")
• Billie Jean King
• Adolf Hitler
• Ozzy Osbourne
• Klaus Barbie
• Gary Hart
• Douglas MacAuthur
• Deng Xiaoping
• Franklin Roosevelt

Soon after he left SNL he co-wrote About Last Night and played the role of Sweetchuck in three of the Police Academy films. Pre-SNL, he also had an uncredited appearance in the opening scene of Scanners. Kazurinsky also appeared on a Season Six episode of Married...with Children where Al Bundy mistakes an angel (played by Kazurinsky) who was murdered by a hit man his family hired for God and creates God's shoes (which turned out to be an invention that the angel created in the 1920s).

Kazurinsky also had a minor role as an alcoholic attorney in the campy 2000 film Poor White Trash which stars Jaime Presley and William Devane.