With the game tied 3-3 late in the third, a Phoenix counterattack saw a blocked shot ricochet into the left circle, where Halpern was waiting all alone.
With Jonathan Quick scrambling following the change of direction, Halpern threw a shot on net that the LA netminder appeared to stop from his back before making another incredible save on the rebound attempt to keep the game tied.
However, at the next stoppage, video review showed Halpern's shot fully cross the goal line before Quick's stick knocked it back into play.
Mike Smith, who finished with 36 saves, stood tall the rest of the way to help the Coyotes claim their second straight win and fourth in the last five games.
Keith Yandle notched a goal and an assist, while Rob Klinkhammer and Mikkel Boedker each scored once for Phoenix.
Marian Gaborik, Alec Martinez and Tanner Pearson all lit the lamp, while Quick stopped 23 shots for the Kings, who entered the game 21-0-0 this season when leading after two periods.