The Kings' worst droughts came to an end Thursday on a night they also pulled out of a stunning free-fall against the Phoenix Coyotes.
Dwight King punctuated the triumph with an empty-net goal that gave him his first career hat trick.
Richards' and Frattin's goals came earlier in the third period, saving the Kings (7-4-0) from great indignity after they seized a 4-0 first-period lead and watched it all vanish.
"Good thing they didn't kick a field goal on the last play of the game," Kings Coach Darryl Sutter cracked.
Phoenix (6-3-2) made it 4-4 by scoring twice in the first 2 minutes 50 seconds of the third period on a deflected power-play goal by Martin Hanzal and a long, low shot by captain Shane Doan that skidded past the right leg pad of Kings goalie Jonathan Quick.
"It went up and down — before we got back to how we play," King said. "You refocus, go back to basics, calm yourself down and get back to playing the game."
The situation was at its most dire when Kings defenseman Jake Muzzin was called for a slashing penalty 41 seconds after Doan's goal.
"It was awful that we let them back in the game, but when a team comes back that dominantly, the leading team [usually] just folds completely," Kings defenseman Drew Doughty said. "We didn't. We stayed strong, got a goal back. And insurance."
Richards keyed the resurgence by stealing the puck from Coyotes forward Radim Vrbata and giving the Kings their third short-handed goal of the season, beating Phoenix goalie Thomas Greiss to his left.
Richards slapped the glass, with emotions of exasperation, celebration and relief all evident, after scoring his 28th career short-handed goal that ties him for most among active players with Tampa Bay's Martin St. Louis.
"That was the turning point," Doughty said. "If they got that next goal, there's no way we're winning that game."
It wasn't supposed to be that tight after Kopitar whipped in a shot 42 seconds into the game to ignite the four-goal first against Phoenix's starting goalie Mike Smith.
Kopitar had gone 26 regular-season games without a goal, his last having been scored on March 25 at Chicago.
In their 11th game of the season, the goal was also the first by a Kings center.
Kopitar then notched his ninth assist on King's first goal 4:17 into the game.
"He goes about his business every night," Sutter said of King. "He doesn't stand out, but it's good to see him get rewarded for goals. He's our most solid left winger night in and night out."
Fraser followed by picking up his first point of the season, pounding a shot that bounced off Smith and was rebounded into the net by Kings forward Jordan Nolan.
King then deflected in a shot from defenseman Robyn Regehr.