Joe Pavelski put San Jose ahead in the first period, but Francois Beauchemin and Matt Beleskey scored in the second to turn the game around and made it six wins out of six by beating the Anaheim Ducks 3-2
Logan Couture scored with 2:45 left to play in the third, then Handzus struck in the shootout to ensure the Sharks kept rolling.
Nathan Horton scored with just over four minutes to play to take the game into overtime and the Boston Bruins beat the New Jersey Devils 2-1 after a shootout to maintain their strong start to the season.
David Clarkson put the Devils ahead in the second period, but Horton intervened just as it looked like the Bruins would suffer their first loss in regulation this year.
Brad Marchand then scored what proved to be the decisive shootout goal and the Eastern Conference leaders improved their record to 5-0-1.
Michael Del Zotto had a goal and an assist as the New York Rangers beat the Philadelphia Flyers 2-1.
Ryan Callahan scored the other Rangers goal but then had to leave the game with what looked like an arm injury.
The Rangers' New York neighbours, the Islanders, also won, with Matt Moulson registering a goal and an assist in a 4-1 victory over the Pittsburgh Penguins.
The Detroit Red Wings beat the Dallas Stars by the same scoreline, with Valtteri Filppula netting twice.
The Tampa Bay Lightning continued their good start to the year as they beat the Florida Panthers 5-2 for their fifth win in six games.
Matt Frantin scored two, including a last-gasp winner in overtime, as the Toronto Maple Leafs beat the Buffalo Sabres 4-3.
The Montreal Candiens were also 4-3 winners, against the Winnipeg Jets, the Ottawa Senators saw off the Washington Capitals 3-2 and the Minnesota Wild got their first win in four with a 3-2 victory over the Columbus Blue Jackets.