Gaborik redirected Anze Kopitar's shot past Jonas Hiller for the winner before being mobbed in the corner by his teammates.
The Ducks appeared poised for a win after 43-year-old Teemu Selanne beat Jonathan Quick between the pads for a go-ahead goal with 11:52 to play in regulation.
But the Kings pulled Quick for a sixth attacker with 1:22 left. Mike Richards took a shot from along the left boards, and Gaborik wristed the rebound past Hiller.
It was L.A.'s fifth straight win.