Zack MacEwen scored twice, Drake Batherson had a goal and two assists, and Jake Sanderson and Chabot each had a goal and an assist for the Senators (2-1-0), who rallied from three two-goal deficits.
Ottawa was without starting goaltender Linus Ullmark because of an undisclosed injury. Mads Sogaard made 12 saves on 16 shots in relief of Anton Forsberg, who allowed three goals on nine shots.
Alex Laferriere and Kevin Fiala each scored twice for the Kings (1-0-2), and Brandt Clarke and Anze Kopitar each had three assists. Kuemper made 33 saves.
"We were up, but I don't think at any point we felt like we were in control of the game," Los Angeles coach Jim Hiller said. "We were pretty opportunistic to be up 4-2, so there was no real shift for me. It was just us trying to get playing the type of game that we needed to get playing. And I don't know that we got that for more than three or four shifts."
Senators defenseman Artem Zub left at 6:30 of the first period because of an upper-body injury sustained on a hit from Los Angeles forward Tanner Jeannot. Coach Travis Green did not have an update on Zub's condition after the game.
Fiala one-timed a pass from Kopitar on the power play to give the Kings a 1-0 lead at 8:12 of the first period. Alex Turcotte's shot deflected in off Trevor Lewis' body to make it 2-0 at 9:07.
Sanderson cut it to 2-1 at 10:25 when he beat Kuemper with a wrist shot over the glove from the slot on the power play.
Adrian Kempe one-timed a pass from Clarke on the power play to make it 3-1 at 5:31 of the second period.
MacEwen made it 3-2 at 10:26, but Laferriere tipped a centering pass from Kyle Burroughs to put Los Angeles ahead 4-2 at 11:16.
"We're up 4-2 in the second period and that's ... I don't care that they come back or whatever," Kempe said. "We have to shut the game down at that point. Unacceptable from our side. Defending poorly, careless, everything wasn't clicking tonight."
Batherson scored on the power play to cut it to 4-3 at 15:13, and Chabot out-waited Kuemper and scored into an open net to tie it 4-4 at 15:32.
"We didn't bring our checking game. That's the bottom line for me," Hiller said. "We didn't have the mindset to check. It turned into a crazy game that we could have won, but there's a reason that we didn't win it."