The NBA scoring champion hit 12-of-20 shots and added eight rebounds to help the Western Conference champions outscore the Heat by 18 points in the second half to draw first blood in the best-of-seven series.
"You know, this is a tough series," said Durant, whose 17 fourth-quarter points slammed the door shut on Miami.
"This level of basketball is the hardest we play, and we just want to take it slow and take it a possession at a time.
"Guys got to just continue to believe in themselves. If we do that as a group, we'll be all right. It took us a couple minutes to get the nervousness out of us."
James, a three-time MVP, scored 30 points on 11-of-24 shooting, while adding nine rebounds, but it was Durant who triggered the Thunder's turnaround.
"They didn't make many mistakes in the fourth quarter," James said of the Thunder's 31-21 advantage in the final period. "I don't think we made many mistakes in the fourth quarter, it's just they made more plays, especially offensively.
"They made more shots. Russell made some big-time shots, KD (Durant) made some big-time shots. So I think that's what it's about."
Durant's tussle with James is widely regarded as a barometer for determining the league's best player, with both also seeking a first championship.
Thunder point guard Russell Westbrook scored 18 of his 27 points in the second half, while adding eight rebounds and 11 assists overall. The only other player scoring in double figures for Oklahoma City was forward Serge Ibaka, who netted 10.
"He's a terrific player. We want him to score," Thunder coach Scott Brooks said of Westbrook. "When he gets to those spots that he can knock those shots down, those are great shots for him.
"Defensively he was solid. When you guard (Dwyane) Wade or LeBron, you have to really exert a lot of effort defensively, and he did that. And he also had enough to finish around the basket."
Miami led by as many as 13 in the opening half but fell apart in the third quarter, and when Westbrook made a three-point play late in the closing seconds to give the Thunder their first lead at 74-73, the Chesapeake Energy Arena erupted.
"They got out in transition and that's their bread and butter, especially here at home," said Heat forward Shane Battier, who scored 17 points but only four in the second half.
"They run so well. For a while we contained them. Second half we made too many careless turnovers."
Game Two is also in Oklahoma City on Thursday before the series shifts to Miami for the next two contests.
"This one is behind us now," Heat coach Erik Spoelstra said. "About halfway through the fourth quarter, we were thinking we were putting ourselves in a position to win.
"And then they just went away. That's what they do, they keep on coming. They're relentless. They beat us at their game and beat us in a game that's very similar to us when we're playing well."
Wade conceded the Thunder "did a great job of turning up defensive pressure" in the second half.
"They just made more plays than us," said Wade, an eight-time All-Star who had 19 points. "They got a couple offensive rebounds that kind of hurt us. They got a couple open shots.
"And then from that point, we were kind of playing behind."