Justin Schultz netted a leveller for the Oilers with 2:37 remaining in regulation, setting up the shootout, and Klinkhammer found the net to end San Jose???s three-game winning streak.
In a thrilling encounter, Jordan Eberle scored twice in regulation for Edmonton while Derek Roy was also on target for the visitors.
Joe Pavelski added a brace for the Sharks, with his second opening up a 4-3 lead at 9:28 of the third period, but Schultz then denied San Jose, sending a wrist shot under goalie Antti Niemi.
Edmonton goaltender Viktor Fasth held the hosts at bay with 33 saves, preventing Pavelski from completing a hat trick in the closing seconds of regulation.
Elsewhere, defenseman Dan Boyle claimed a fortunate goal for the New York Rangers in their 6-3 win over Florida Panthers.
Boyle sent a seemingly harmless fling towards the net that took a big deflection off the stick of Panthers defenceman Willie Mitchell before slipping past goaltender Robert Luongo.??