The Sharks' longest winning streak of the season was snapped at five games. It was the final contest to be played before the NHL's All-Star break.
Stoll's goal came in the fourth round of the shootout.
Jamie Quick was perfect in the shootout after stopping 22-of-24 shots in regulation and overtime. Quick won his third straight start after losing four in a row.
Ryan Smyth opened the scoring for Los Angeles with 6:25 to play in the second period, but before the period was over San Jose had taken a 2-1 lead on goals from Ryane Clowe and Devin Setoguchi.
Alexei Ponikarovsky produced the lone goal of the third period to tie it.
Antti Niemi made 18 saves in taking the loss.