The 39-year-old Clouston, promoted from the AHL's Binghamton Senators, took over after Craig Hartsburg was fired Sunday night. He is the team's fourth coach since Ottawa lost in five games to Anaheim in the 2007 finals.
"[Clouston] did a good job behind the bench getting us fired up," forward Jason Spezza said. "It's unfortunate we didn't score a goal. It's going to be a learning process for us here for the next couple of weeks."
The Senators (17-25-7) have lost three straight. Ottawa also lost 1-0 at Columbus on Friday night.
"We've got to do a better job of getting rebounds and getting goals. That's what's missing," captain Daniel Alfredsson said.
The Senators are the lowest-scoring team in the NHL with just 116 goals in 49 games.
Jonathan Quick made 29 saves to record his third shutout of the season and improve to 4-1-0 in his last five starts. It was the first time he had faced the Senators.
"You had to expect them to come out with a bit of jump with the new coach there," Quick said. "They had a good first period and in the second period you could tell they were really trying to pour it on put the defense played great. The difference was a greasy goal at the end there."
Handzus picked up the puck behind the Senators net and jammed it in from the side at 17:21. Wayne Simmonds and Brad Richardson picked up assists on Handzus' ninth goal of the season.
"I was just trying to get the puck to the net and hope for a rebound," Handzus said.
"That was a huge win for us."
Brian Elliott stopped 25 shots for Ottawa, coming very close to his first career shutout.
"Their best player was their goaltender tonight in Quick. He was very, very solid," Clouston said.
"We had some opportunities and he's very athletic. We just couldn't capitalize on our chances."
Kings D Denis Gauthier served the first game of a five-game suspension he received for a hit on Montreal's Josh Gorges on Saturday. ... Kings D Matt Greene played in his 200th game. ... The Senators haven't scored against Los Angeles since December 2005. They were shut out 2-0 by the Kings in their only meeting last season and didn't play each other in 2006-07.
Copyright by STATS LLC and The Associated Press
Arena: Canadian Tire Centre Location: Ottawa, Ontario
Referees: Eric Furlatt, Dean Morton Linesmen: Brad Kovachik, Derek Nansen
Attendance: 18,054 (97.6% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
No scoring this period
No scoring this period
Michal Handzus (9)
Assists: Wayne Simmonds, Brad Richardson