The Red Wings, who have won two straight games, are one point ahead of Columbus and one behind seventh-place Minnesota. All three teams have two games remaining in the regular season.
"We're trying to crawl our way back into the playoffs," Detroit's Jimmy Howard said after a 23-save effort.
Detroit has been in every NHL postseason since 1991 in what has become the longest active run in North American professional sports.
"Everybody is talking about 21 years the Detroit Red Wings have made the playoffs," Kings coach Darryl Sutter said. "What's the biggest accomplishment is what they've done since the salary cap."
The Sharks and Red Wings are the only teams who have been in the playoffs every season since the 2004-05 lockout, which led to the league's salary cap.
While San Jose is in the postseason again, Detroit still has work to do to earn a spot.
The 10th-place Dallas Stars, who are four points behind Detroit, host Columbus on Thursday and Detroit on Saturday. The Red Wings will also face the banged-up and already eliminated Nashville Predators on Thursday at home.
Detroit's win eliminated Phoenix from the playoff race.
Howard said the team has gotten used to playing with a sense of urgency toward the end of the lockout-shortened, 48-game schedule.
"They've all felt like a playoff game here for a while," Howard said. "So, it's just two more here, and hopefully get the job done."
The defending Stanley Cup champion Kings are trying to finish fourth to earn home-ice advantage in the first round of the playoffs. They will host San Jose, which also has a chance to be fourth, on Saturday.
With more at stake, Detroit seemed to skate and hit harder. The Kings weren't surprised.
"We knew they were going to come out strong," Los Angeles center Anze Kopitar said. "They're fighting to make the playoffs."
Sutter disagreed with the decision.
"There was no intent," Sutter said. "The player ran into his elbow."
Brown ranks second on the team with 18 goals and fifth with 29 points.
"You don't replace Brown in your lineup," Sutter said. "I look at it another way. He's got two games off to get some rest for the playoffs."
Los Angeles' Jonathan Quick made 33 saves, including 13 in the first period when the Kings recorded just four shots on Howard.
The Kings had a power play midway through the third period, while trailing by a goal, but failed to take advantage.
Soon after the penalty was killed, Detroit went on a power play. A fan at Joe Louis Arena celebrated by throwing an octopus on the ice in what has become a playoff tradition in the Motor City.
Franzen flipped the puck past Quick after defenseman Matt Greene failed to move the burly player known as "The Mule" away from the front of the crease. Babcock said Franzen wasn't able to finish the game, declining to give details, but hoped he would be healthy enough to play against the Predators.
The NBA's San Antonio Spurs are in a 16th straight postseason, the second longest active streak in North American pro sports. ... Chris Ilitch, president and CEO of Ilitch Holdings, which owns the Red Wings, said progress is being made on a new arena for the team. ... Suter said F Dustin Penner was a healthy scratch.