Ryan O'Reilly's shorthanded goal 45 seconds into overtime capped Colorado's rally from a two-goal deficit in the third period at Pepsi Center.
O'Reilly spun around a defender near the net before beating Martin Jones with a backhander for the winner.
"That looked like a Peter [Forsberg] move for sure," coach Patrick Roy said.
Dwight King's power-play goal 38 seconds into the third period put L.A. up 3-1. But Steve Downie roofed a power-play goal at 3:20 and O'Reilly forced overtime by pumping his own rebound behind Martin Jones.
Colorado's Andre Benoit beat Ben Scrivens 26 seconds after the opening faceoff to give the Avalanche a quick lead. Justin Williams' power-play goal 25 seconds later got the Kings even, and Los Angeles went in front at 3:10 when Denver native Nick Shore one-timed Williams' pass behind Slava Voynov.
It was Colorado's first victory under Roy, who was hired as coach this summer.
"Our guys really want our fans to be proud of this team," Roy said.