0350 GMT August 18, 2022
The 36-year-old point guard made the crucial difference in Game 1, shooting 12 of 19 from the floor, BBC Sport reported.
Devin Booker scored 27 points, while DeAndre Ayton added 22 points and 19 rebounds.
It is the Suns' first Finals appearance since 1993 as they hunt a first title.
The Bucks welcomed Giannis Antetokounmpo back after he missed two Eastern Conference games with a hyper-extended knee. He scored 20 points and 17 rebounds in 35 minutes.
Paul, who said his team-mates anticipated Antetokounmpo's return, added that the Suns would take nothing for granted in the series.
"One thing we are going to be is prepared for every game," he said.
"We always talk about when preparation meets opportunity. We played against these guys twice.
"We prepared for Giannis or not Giannis. We're sure they're going to make adjustments going into game two, and we'll probably do the same."
Antetokounmpo had feared a lengthy recovery period and so admitted he was relieved to be able to play in the opening game of the series.
"I'm happy. I thought I was going to be out for a year so I'm happy that two games later, I'm back," he said.
"I'm trying my best not to make it about my knee, but my knee felt good. At the end of the day, I'm out there, I feel good. I don't feel pain; I can run, I can jump, I can set screens and I can rebound the ball."