Bjornson was educated at the Kelvin Technical High School, and worked as an accountant and service manager. He was an officer of Lavoie Motors Ltd. in Lac du Bonnet. He served for fifteen years with Ford Motor Company of Canada, and Ford Motor Company of India. During World War 2, he served as a technical advisor to the government of India.
He was first elected to the Manitoba legislature in the 1959 provincial election, defeating Liberal-Progressive John Ateah by 85 votes in Lac du Bonnet. He was re-elected over Ateah by a greater margin in the 1962 election, and defeated Liberal James Desilets by 80 votes in the 1966 election. He served as a backbench supporter of Premiers Dufferin Roblin and Walter Weir during his time in office.
Bjornson did not run for re-election in the 1969 election, after redistribution significantly changed the Lac du Bonnet riding.