It took place on March 3rd 1981. Independent Bernie Sanders had 4,330 votes. Democratic Gordon Paquette had 4,320 votes.
https://en.wikipedia.org/wiki/1981_Burlington_mayoral_electionUntil that time, Vermont had voted D for president only in 1964, but there was no republican candidate in the 1981 Burlington mayoral election. Bernie Sanders ran against a democrat. Five months earlier, Jimmy Carter narrowly won the Chittenden County: he had 40.4%, Reagan had 39.0%, Anderson had 17.9%. Since Chittenden County is not only Burlington, we can imagine that Carter had a higher margin in Burlington. Even though, the vote for Reagan was significant, and then, there was a Sanders vs Democratic race for mayor.
Here we can see the interview with Bernie Sanders soon after his victory. He was 39 and he had the same face he has now, the only difference is the hair. It is weird to listen to "the conservative state Vermont" in 2020
https://www.youtube.com/watch?v=OraxqbUjpHw&t=89s