Scott Thornton and Michael Cammalleri scored 2½ minutes apart in the second period for the Kings, who avoided finishing last by beating their Pacific Division rivals and snapping a six-game losing streak. Jaroslav Modry and Dustin Brown each had two assists.
Mike Zigomanis and Nick Boynton scored for the Coyotes, who will finish their season Sunday against Vancouver. They lost their final seven road games and finished 13-26-2 away from home.
Los Angeles finished 27-41-14 in its first season under coach Marc Crawford. Of the 13 coaches that made it all the way through their first season with the Kings, Crawford is the fourth with at least 40 losses - joining Don Perry, Larry Robinson and Larry Regan.
The Kings missed the playoffs for the fourth straight season, matching the longest drought in the franchise's 40-year history. They had 68 points, their lowest total in a regulation season since 1996-97, when they had 67.
Jeremy Roenick was sent off for hooking with 17:56 to play and Derek Morris followed him into the penalty box on a tripping infraction 38 seconds later. The Kings capitalized with 49 seconds left on the 5-on-3 advantage, as Kopitar converted Alexander Frolov's pass through the crease.
Los Angeles spotted Phoenix a 2-0 lead before Thornton scored his fifth career short-handed goal at 11:43 of the second period, just nine seconds after Kopitar received a hooking penalty. Cammalleri got the equalizer on a short wrist shot from the right circle past Curtis Joseph at 14:12 of the period - the moment that Niko Kapanen's roughing penalty expired. It was his 34th goal, one fewer than Frolov for the team high.
Kings heralded rookie defenceman Jack Johnson got into his first NHL fight when he went toe-to-toe with Daniel Carcillo at 15:26 of the first period. Less than 2½ minutes later, Zigomanis opened the scoring with his 14th goal.
Kings defenceman Jamie Heward was serving a slashing penalty when Boynton beat Mathieu Garon high to the stick side with a one-timer from the slot at 7:10 of the second period for a 2-0 lead.
It was only the second goal in 58 games this season for the Coyotes defenceman, whose other came March 29 against Colorado.
Notes: Garon finished with a 13-10-6 record. This is the sixth time in Kings history that none of their goaltenders had more than 13 wins . . . It was the third time the Kings went through a season without any player recording a hat trick (also 1970-71 and 2003-04) . . . Los Angeles was 16-16-9 at home. They had nine sellouts and averaged 16,473, the second-lowest figure in their seven seasons at the Staples Center . . . This is the first time since 1985-86 that the Kings failed to win more than two consecutive games. It was also the first time they have played an entire season without getting shut out at home since 1995-96.