The Clippers defeated rivals the Los Angeles Lakers and then went to the Golden State Warriors and won to begin the campaign, but that run ended with a 130-122 defeat.
Star signing Leonard had been the key in each of their prior victories, and he had 27 points, eight rebounds and 10 assists as the Clippers went down on Saturday.
Rivers revealed Leonard, whose playing time was restricted in the regular season last time out with the Toronto Raptors, could have played 35 minutes if he had thought the game was within their grasp.
"He got hot and made a couple of shots. We tried to keep him in that 30-minute range, 35 – we could have gone to 35 if we'd wanted to," Rivers said.
"Honestly, I didn't think we were going to win the game. We could have kept him in, but 35 is a lot of minutes. We could play him [for] that if we want; I just chose not to.
"I'm not going to keep him out there when we're not playing right as a group. Why keep him out there?"
He added: "We never made a run. It's called a fake run. It's what you do in the NBA.
"You're down by 12, you cut it to seven, everybody thinks you made a run, and then they make a wide-open three – like they've been doing all game – and the run's over. I never thought we were a threat."
Giving an assessment of the game as a whole, Rivers paid tribute to the Suns.
"Listen, even if we had won the game, we had no business winning that game," he said. "They just played so much harder, they were so much more committed to what they were doing.
"In every aspect of the game, I thought they were better. Monty [Williams, Suns coach] did a better job. They just really deserved to win the game.
"I thought offensively – we had, what, eight assists at half-time – we've been a ball-moving team and we didn't move the ball. But give them credit – I thought a lot of that was clearly their ball pressure.
"I thought [Jevon] Carter did an amazing job defensively. He just put pressure on everybody. That's what Proviso East kids do, for the record, and he did it. He was absolutely fabulous. I thought [Devin] Booker kept his composure.
"There were a lot of fouls called. We got a lot of them in the first half and you know the old human nature thing: in the second half, it's going to get even somehow. We had a penalty within two minutes. That was troublesome.
"But other than that, we deserved to lose. We really did. Give them credit."