Penner, who had an assist in regulation, pounced on a rebound in the slot and had a clear lane to beat Mike Smith at 17:42 of the extra period to put the Kings in the Stanley Cup finals for the first time since 1993.
Los Angeles, the No. 8 seed in the West, will try meet the winner of the Eastern Conference finals. The New York Rangers and New Jersey Devils are tied 2-2 in their series, with Game 5 set for Wednesday in New York.
Anze Kopitar scored a short-handed goal for Los Angeles in the first period. Drew Doughty and Mike Richards added goals in the second period.
Taylor Pyatt opened the scoring for Phoenix in the first period and assisted on Keith Yandle's goal late in the second that tied the score 3-3. Marc-Antoine Pouliot accounted for the Coyotes' other goal.
Jonathan Quick had 38 saves on 41 shots for LA. Smith stopped 47 of the 51 shots he faced in goal for the Coyotes.