Brown scored during a 5-on-3 advantage and Lewis provided the winner on a loose puck with 1:21 left, sending Staples Center into shocked celebration. The defending Stanley Cup champions won their 12th straight home game since March and their sixth straight postseason game.
Marc-Edouard Vlasic scored his first career playoff goal with 11:04 to play for the Sharks, who were doomed by two late penalties after rallying from an early two-goal deficit.
Jonathan Quick made 28 saves for the Kings, and Antti Niemi stopped 27 shots for the Sharks.
Game 3 is Saturday night at the Shark Tank.
Patrick Marleau and Brad Stuart scored second-period goals for the Sharks, who largely dominated the first two periods with another barrage of superior speed and skill. Los Angeles got rolling in the third period - and with two late goals, the Kings landed one of the most amazing victories in their relatively thin playoff history.
Jeff Carter scored on the Kings' first shot and added an assist on Brown's goal in front of Niemi. Drew Doughty scored an early power-play goal for Los Angeles.
After Vlasic's score on a rebound, the Sharks seemed to be cruising to a win - until Stuart was whistled for tripping with 2:41 to play. Just 22 seconds later, Vlasic joined him in the box when he shot the puck over the glass, although Vlasic claimed the puck had deflected off Carter.
The Kings scored a minute later, with Brown pounding home his second goal of the postseason. Lewis got the winner shortly after the ensuing faceoff, finding a loose puck on the back side of a scrum in front of Niemi. Lewis hadn't scored in the postseason, but the depth forward scored two goals in the Kings' Cup-clinching win over New Jersey last June.
The Kings played without veteran center Jarret Stoll, who didn't return to Game 1 after a hit from San Jose's Raffi Torres late in the second period. Torres was suspended for the rest of the series Thursday by the NHL's Department of Player Safety for his illegal check to the head after a hearing in New York.
Brad Richardson contributed a key power-play assist while filling in for Stoll.
Los Angeles has won six straight home playoff games dating to last season's Stanley Cup clincher, but none was more surprising than this victory. The Sharks appeared to be sure Quick couldn't steal another win after the Kings took the opener despite getting outshot 35-20.
Quick's streak of more than 125 minutes of shutout hockey ended in the second period with two quick goals. Vlasic then gave San Jose its first lead with a rebound goal on Stuart's shot off the end boards midway through the period, capping a lengthy scramble near Quick's net. Vlasic didn't score a goal in his first 73 career playoff games.
The Kings were forced to make several changes in the absence of Stoll, the third-line center and faceoff specialist who excels on special teams. Stoll was replaced by Richardson, who played 13 postseason games during the Kings' championship run last summer.
The Sharks replaced Torres by juggling their lineup, promoting Joe Pavelski to the second line and using several new combinations. Tim Kennedy was in uniform for just his second game of the postseason for San Jose, which also played without injured forward Marty Havlat.
NOTES: Sharks F Adam Burish sat out his second straight game after playing in all four games against Vancouver. He was replaced again by Bracken Kearns. Havlat hasn't played since the playoff opener in Vancouver while nursing a lower-body injury. ... Richardson hadn't played since the Kings' playoff opener against St. Louis on April 30. He appeared in just 16 regular-season games. ... Quick hadn't allowed a goal since early in the second period of the Kings' series-clinching win over St. Louis six days earlier.