With the Senators clinging to a 2-1 lead early in the third period, Zack Smith stripped the puck from Toronto's Matthew Lombardi and chipped it ahead to Daugavins, whose floater eluded Jonas Gustavsson for a goal.
Daugavins, a 2006 third-round Senators draft pick from Russia, was playing in only his fifth NHL game.
Colin Greening and Chris Neil also scored while 20-year-old rookie goalie Robin Lehner turned aside 23-of-25 shots to earn the victory.
Clarke MacArthur netted both goals for the Maple Leafs, who lost for just the third time in 11 games to start the season.
Gustavsson ended with 27 saves for Toronto.
Celebrity Families of 2014 [PHOTOS]