Without Paul George, the Clippers won 112-102 against a Lakers team containing LeBron James and Anthony Davis at Staples Center.
Leonard produced an influential performance, as a game-high 30 points came with six rebounds, five assists, two steals and a block.
He was heavily booed by Lakers fans after turning down the chance to join the team in favour of joining their city rivals.
But Leonard, who won the NBA title with the Toronto Raptors last year, provided the perfect answer, hitting seven straight shots at one stage as the Clippers scored 40 points in the second quarter.
"All those emotions went through me again," Leonard said of his Clippers debut back in his hometown.
"Once I'm playing the game, I'm locked in. But before the game last night there were a lot of emotions, knowing my family was going to be here and that I'm going to be playing here at Staples, my hometown – it's just amazing.
"It was great, a lot of emotions, game one. I'm just happy we came out and got a victory. We played a great practice the last week, everyone competing and pushing everybody's limits and we had some carry over.
"I just made shots pretty much. I felt like I got to my spots early, missed some little chippies but started making shots. That's what happens."
Lakers fans targeted Leonard with boos when the lineups were announced and again as he attempted to address the Clippers' fans before the game.
"I just heard a loud noise," he said. "I was just trying to be as loud as I can. I wasn't sure if the Clipper fans were being loud or if it was the Laker fans.
"I didn't pay attention to it too much. I was just trying to let the fans know the appreciation we have for them for coming out."
Clippers coach Doc Rivers was impressed by Leonard's first outing, while also recognising the importance of the bench, who outscored the Lakers' second unit 60-19.
"He talks with his game," Rivers said of Leonard. "I think that's the way it should be. It's a great sign of leadership.
"He was intense. You could feel the intensity, there's no doubt about that. And this bench is going to be better this year than even last year."
Leonard, who was 10-of-19 shooting in 32 minutes, added: "These guys were a playoff team last year, they fight hard every single night. We are not going to stop just because I'm here or Paul's here.
"We're missing a couple more pieces still, with Paul and a great defender Rodney [McGruder] too."
The Clippers have another high-profile game up next, on the road against the Golden State Warriors on Thursday.