Los Angeles goalie Jonathan Quick improved his shootout record to a perfect 10-0 this season turning aside two of three shooters.
Stoll opened the scoring in the first period to stake the Kings to a 1-0 lead, but the Coyotes roared back with goals by Lee Stempniak and Taylor Pyatt before the end of the period.
Kyle Clifford scored the equalizer late in the second period, redirecting a perfect pass by Wayne Simmonds to set up the dramatic conclusion.
(Reporting by Mike Mouat in Windsor, Ontario. Editing by Alastair Himmer.)