It was the ninth shutout of Varlamov's five-year career, the first three having been spent with Washington.
After giving up four goals in his first start of the season, Varlamov turned aside all but one of 29 shots on Tuesday in a win over the defending Stanley Cup champion Los Angeles Kings.
Columbus has lost three in a row after capturing its first game of the season.
It remained a 2-0 game until John Mitchell scored 4:38 into the third. Duchene added the final tally with 1:18 to play.
All four goals came against Sergei Bobrovsky, who faced 28 shots.