Tsonga needed just around two hours to knock off Dancevic 6-4, 6-4, 6-1 and give France an unbeatable 3-1 lead in the best-of-five tie with Canada.
The French, who carried a 2-1 lead into Sunday, will now host a quarterfinal tie with the United States in April. The Americans completed a surprising 5-0 sweep of Switzerland earlier in the day.
Tsonga's scheduled opponent, Milos Raonic, pulled out of the match because of knee pain he felt following a doubles loss on Saturday.
Taking his place, Dancevic made 27 unforced errors -- and had the same number of winners -- while Tsonga fired nine aces and 41 winners against 20 unforced errors.
France owns nine Davis Cup titles, the last coming in 2001, while Canada has never reached a World Group final. France improved to 2-0 all-time against the Canadians, with the previous victory coming in Paris in 1966.