Born in Kennett, Missouri, he got his start as a sports cartoonist for the Tampa Daily Times. While playing golf in the area, "Barney Google" creator Billy DeBeck noticed Lasswell's work and hired the 17-year-old as an assistant. After DeBeck's death of cancer in 1942, Lasswell took over the strip. Originally a sports- and urban-oriented strip, the cartoon introduced Google's hillbilly cousin Snuffy Smith in 1934. Under Lasswell's tenure, Google was gradually phased out and the strip's emphasis shifted to Smith, drawing on his rural upbringing for the stories. Lasswell also introduced his own characters, including Tater, Elviney Barlow, Parson Tuttle and Ol' Doc Pritchart. For his work on the strip, he received the National Cartoonist Society Humor Comic Strip Award in 1963 and its Reuben Award, which had originally been named after DeBeck, that same year. He also received their Elzie Segar Award in 1984 and 1994.
Despite his rural background and the nature of his work, Lasswell was a prolific inventor and early adopter of new technology. He was one of the first cartoonists to email his strips to his syndicate, King Features Syndicate, and to employ computer-generated lettering. A member of the American Society of Agricultural Engineers, he patented a citrus fruit harvester.