Patrick Fischer and Jean-Jacques Aeschlimann scored the goals as the Swiss, who were already eliminated from the final round, finished their disappointing Olympic campaign on a high.
Marcel Jenni and Reto Von Arx were missing from the Switzerland line-up after they were sent home in disgrace by coach Ron Krueger for violating a team curfew.
Fischer opened the scoring at 16:52 in the opening period, but Belarus levelled through Vadim Bekbulatov at 3:42 in the middle period.
Aeschlimann scored the winning goal at 9:11 in the second period to seal Switzerland's first win in the competition.
Despite the defeat, Belarus top the group with four points, after winning two of their three games.
In the other Group B match, Roman Salnikov assisted on three goals as Ukraine beat France in a match-up of two teams unable to make it to the final round.
Dmitri Khristych, who plays for the Washington Capitals in the NHL, and Igor Chybirev scored power play goals for Ukraine in the opening period.
Vadym Shakraychuk and Oleksiy Ponikarovsky added goals in the second period to ice the win.
Former NHL player Phillipe Bozon, appearing in his fourth Olympics, scored both goals in the middle period for France.
Ukraine finish second in the group with four points, with France finishing bottom after one draw and no wins from their three games.