With an unusual offensive burst, the Kings proved their captain correct and headed back to San Jose with a flicker of hope.
Brown had a goal and an assist, and Tyler Toffoli scored on his 22nd birthday during a three-goal second period as the Kings dodged a series sweep with a win that could plant a seed of doubt in their dominant California rivals' minds.
"There's a lot of pride in here," Brown said. "We've been through a lot as a group, and this is one more challenge. They're still in command of the series, but we're not going to stop trying."
Game 5 is Saturday in San Jose.
The defense-first Kings were outscored 13-5 in the first two games at the Shark Tank before losing Game 3 in overtime, but the back-to-back Western Conference finalists finally produced a strong two-way performance when they faced an early playoff exit.
Jonathan Quick made 36 saves for Los Angeles, which chased Sharks goalie Antti Niemi on Gaborik's beautiful goal early in the third period. The final minute of the rivals' third postseason series in four years was marred by more goonery and shenanigans, including a wild fight between San Jose's Brent Burns and Los Angeles' Robyn Regehr after the final buzzer.
"We're trying to just make it a series now," said Williams, who got his first two goals of the postseason after dropping off the Kings' top line. "It's obviously not like a Game 7, but it was do or die for us, so we had to lay that on the line. It wasn't your typical Kings game."
After three scoreless games, Brown had a key early assist and an empty-net goal for Los Angeles, which got points from 12 players in Game 4 after just 10 Kings scored in the first three games combined.
Gaborik has three goals in his first four playoff games with the Kings, who are getting what they sought when they acquired the Slovak star from Columbus last month.
James Sheppard, Matt Nieto and Joe Pavelski scored for the Sharks, who couldn't finish off the second playoff sweep in franchise history. San Jose again played its swarming offensive game for long stretches but got none of the fortunate deflections and hustle-created luck of its Game 3 win.
"I thought we had some intensity to our game and I thought we were as desperate as they were, but we didn't execute anywhere near the way they did," Sharks coach Todd McLellan said. "It's quite easy to pinpoint some of the mistakes that we made, and for us to have success moving forward, that has to get cleaned up quickly."
Los Angeles shuffled its top line for Game 4, putting Gaborik and Brown, who hadn't scored in the series, alongside center Anze Kopitar. The changes paid off just 4:08 in when Brown pushed the puck down the wing and flung it in front for Gaborik.
San Jose evened it with 7.3 seconds left in the period on Sheppard's first goal in 21 career playoff games.
After Williams put the Kings back ahead, the Sharks pulled even again on Nieto's chip shot off defenseman Alec Martinez's shoulder.
Instead of deflating, the Kings kept pushing.
After Williams reclaimed the lead by tucking in a long rebound off the end boards, Toffoli swept home a loose puck that deflected off Andrew Desjardins' glove in the final minute of the period, sending Los Angeles into the third with a standing ovation.
Alex Stalock made several big saves in his NHL playoff debut after relieving Niemi, setting up a difficult decision for McLellan before Game 5.
"I don't necessarily think the scoreboard did us justice," Sharks captain Joe Thornton said. "I think we played a pretty solid game. I think they just got some more of the puck bounces tonight, but we showed a lot of fight. We came down here and did what we're supposed to do. Now we have to go home and do our job."
Williams thought he had completed a hat trick with 8:50 to play, but his goal was waved off due to a cross-checking penalty on Jarret Stoll in front of the Sharks' net an instant earlier. Pavelski scored on a rebound moments later during the power play, but Quick kept the Sharks scoreless the rest of the way.
Nieto, a Long Beach native, scored in both games at Staples Center against his favorite childhood team. ... Sheppard was the 12th Sharks player to score a goal in the series. ... The Kings haven't been swept in a playoff series since 2000. San Jose earned its only sweep in the first round against Vancouver last spring. ... Pau Gasol and Larry David watched the game from seats on the glass.