The Dallas captain collected a back pass from Mike Ribeiro and eluded three Kings to score his 26th goal of the season for the Stars, who have won five of their past six and haven't lost in regulation since Feb. 22.
Ribeiro had a goal and two assists, and Kari Lehtonen made 20 saves as the Stars wrapped up a four-game trip against the entire Pacific Division with a strong finish in their third California road game in four nights.
Trevor Daley and Jamie Benn scored tying goals for Dallas in the third period, with Benn getting a short-handed breakaway goal with 14:40 to play.
The Stars had lost five straight before their current surge, which has them in fifth place in the muddled Western Conference standings.
"It showed huge character for us to come out and get that two points," Morrow said of the Stars' three rallies from one-goal deficits. "This group has been good about that all year. Last year and in years past, it was something we wouldn't be mentally tough enough to overcome. But now we just wipe those things away and try to steal the momentum on the next shift."
Los Angeles never trailed in regulation, but Dallas tied it three times. Benn's breakaway goal erased a lead created by Clifford, a hardworking rookie who turned 20 in January.
"The one-goal lead going into the third gave us a good position over a team in our division," Los Angeles coach Terry Murray said. "We should be able to shut that game down. We didn't manage the puck. We made mistakes and turned the puck over. That's something we need to clean up right away."
Jonathan Bernier made 18 saves, Anze Kopitar had two assists and Williams earned his 400th career point with an assist for the Kings, still stuck in eighth place in the West. Up next is a four-game road trip, their final significant stretch away from Staples Center during the chase for their second straight playoff berth.
Mitchell scored his third goal of the season in the opening minutes, wiring a low shot from the point that deflected right through defenseman Stephane Robidas and Lehtonen. The goal was Mitchell's first at Staples Center during his first season with the Kings.
Dallas didn't get its first shot on goal until 9:17 in, but Ribeiro tied it in the second period when he joined an odd-man rush with no defenders near him, firing home his 13th goal from close range.
Just 35 seconds later, Williams fired a long shot and then split two defensemen to collect the long rebound allowed by Lehtonen, banging it home on his second hack for his 21st goal.
"We're a team that if we're going to make the playoffs, we're going to need to lock games like these down," said Williams, who earned his 400th career point with an assist. "We missed an opportunity to jump ahead of these guys in the standings. On the other side, we're going to make sure that that point isn't going to stop us from making the playoffs. We have a big road trip coming up, so we're going to have to have a big push."
Daley evened it at 2 by sneaking into the Kings' defense for his first goal since Jan. 15 early in the third period. Clifford answered 44 seconds later, putting a perfectly placed shot in the far corner above Lehtonen's glove for his first goal in 20 games.
But Kings defenseman Drew Doughty's pass from the point during a Los Angeles power play was intercepted by Benn, who skated in untouched for his 17th goal.
"It took a lot of grit," Dallas coach Marc Crawford said. "It's funny, because you usually don't get many opportunities against the Kings because they play solidly defensively. That's been our secret lately, too. We haven't given up many opportunities, but there were a few mistakes made tonight on both sides, and when you made the mistakes, they ended up being on the sticks of guys that had something going."
Material from wire services and team media was used in this report.