Steve Kerr sounded like an NFL head coach before Monday night’s game at Oracle Arena, preaching “ball security” as if it were the gospel and his starting running back had betrayed him with a costly fumble.

Kerr was asked if he noticed anything troubling while reviewing video of his team’s mostly impressive Game 1 victory over San Antonio on Saturday. He quickly offered a familiar lament: He was less than thrilled with the nonchalant way the Warriors handled the basketball, even if they won by 21 points.

“I thought we got very careless with the ball in transition,” Kerr said. “I’m interested in our ball security. To me, that’s what this series is about.”

Then, as if to illustrate Kerr’s point, the Warriors committed 11 turnovers in the first half of Game 2 on Monday night and fell behind by six points. They treated the ball with more respect in the second half — and not coincidentally surged to a 116-101 victory.

The game reversed course in the third quarter, when the Warriors outscored the Spurs 33-22 to seize a five-point lead. And how many turnovers did Golden State have in the third quarter? Two.

And the Warriors also committed just two in the fourth quarter, when they pulled away.

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“They were tremendous defensively in the first half,” Kerr said of the Spurs. “They got after us and took away everything we were trying to do. ... We got disjointed, but we did a much better job of moving the ball.”

Let’s be realistic: The Warriors, the league’s highest-scoring team in the regular season (113.5 points per game), are borderline impossible to stop when they take care of the ball. Kevin Durant and Klay Thompson are extraordinary shooters, but turnovers obviously prevent them from even attempting a shot.

This is not a new problem for the Warriors, of course. They averaged 15 turnovers per game during the regular season, tied with the Hawks and Suns for third most in the NBA. Only the Lakers and 76ers averaged more.

Not surprisingly, three of those other four teams did not make the playoffs.

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The Warriors hit their average in Game 1, an issue largely overlooked because they rolled to victory. But Kerr noticed, clearly, and so did his counterpart, San Antonio head coach Gregg Popovich.

Popovich had his Spurs aggressively pressure the ball in Game 2, especially in the first half. They even picked up full-court occasionally, a rare tactic in the NBA.

“It’s the playoffs,” Warriors guard Shaun Livingston said. “They’re fighting for their lives.”

This cranked-up pressure also made sense, given the state of the Warriors. They’re prone to turnovers, anyway, and their two-time MVP point guard, Stephen Curry, is sidelined for the series with a sprained left MCL.

Curry’s absence prompted Kerr to slip Andre Iguodala into the starting lineup in the first two games, at point guard. The move largely worked, but it also left the Warriors without a traditional ball handler; Iguodala and Draymond Green can initiate the offense, but they really are forwards.

The Warriors spread the sloppiness in the first half. Some snapshots: Livingston made a careless pass, Iguodala lost the ball off the dribble and JaVale McGee curiously put the ball on the floor after catching a pass on the baseline.

They curbed such mistakes in the second half, and that made all the difference.

Ron Kroichick is a San Francisco Chronicle staff writer. Email: rkroichick@sfchronicle.com Twitter: @ronkroichick