John Tavares, Josh Bailey and Anders Lee also scored to give the Islanders the lead after they trailed 2-0. Thomas Greiss stopped 26 shots to help New York win for just the second time in seven games (2-4-1) and improve to 10-2-2 at home.
Kopitar tied the score 3-3 with 13 seconds remaining in regulation when he knocked the puck past Greiss after a scramble in the crease. Kopitar's team-leading 17th goal came after Lee had given New York a 3-2 lead with 3:16 left.
Lee's goal completed a three-goal rally by the Islanders after the Kings scored twice in the first period.
Bailey narrowed the margin to 2-1 with 50 seconds to go in the opening period with his ninth of the season with assists to Tavares and defenseman Calvin de Haan. Bailey, tied for the team lead with Tavares with 41 points, has points in five straight games -- four goals and five assists, including his first career hat trick against Columbus on Thursday.
Tavares tied the score with a power-play goal at 4:15 of the third after Kings defenseman Jake Muzzin was penalized for elbowing Cal Clutterbuck at 3:50. Tavares whipped the puck past Kuemper for his 19th with assists to Lee and Bailey.
Fantenberg opened the scoring with his second of the season at 12:56 of the first period. The 26-year-old rookie defenseman from Sweden blasted a shot from the middle of the blueline past Greiss.
Toffoli then made it 2-0 on the power-play with 2:49 left in the opening period with his 15th of the season. It gave the Kings a power-play goal in five straight games.
The Islanders held a 14-10 shots on goal advantage in the opening period.
The Islanders outshot the Kings 10-8 in the scoreless middle period that included two power-play chances for the Islanders. Kuemper's best saves were on successful point shots from Tavares and Ryan Pulock.
The Kings were coming off a 4-2 loss to the Rangers at Madison Square Garden on Friday after dropping a 5-1 decision to the Devils on Tuesday that ended their long winning streak.
Kuemper started for the Kings after Quick played against the Rangers.
F Steve Bernier, recalled from Bridgeport on Friday, made his season debut for the Islanders while skating on a line with Anthony Beauvillier and Chimera. Bernier had 10 goals and four assists in 21 games for Bridgeport. The 32-year-old winger played 24 games for the Islanders in 2015-16 and spent all of last season with the Sound Tigers. Bernier was recalled when Josh Ho-Sang was sent down. ... Los Angeles won the first meeting between the teams on Oct. 17 in LA, 3-2. ... The Islanders have allowed the first goal in 22 of their first 33 games. ... New York scratched forwards Alan Quine and Shane Prince plus defenseman Dennis Seidenberg. ... The Kings scratched forwards Andy Andreoff and Jonny Brodzinski and defenseman Kurtis MacDermid.
Kings: At Philadelphia on Monday night to conclude a four-game trip.
Islanders: Host Detroit on Tuesday night.