Like many big-city mayors in recent days, Ms. Bottoms, a first-term Democrat, had little time to think and everything to lose. She understood the outrage driving the protests. But if she failed to restore order, the city could burn.

She needed to find the perfect words.

The mayor’s speech that night lasted a little more than four minutes. She aimed her remarks directly at the rioters. As an African-American mother of four, she told them, “You’re not going to outconcern me and outcare about where we are in America. I wear this each and every day. I pray over my children each and every day.”

She reminded them of Atlanta’s long tradition of black leadership and the legacy of Martin Luther King Jr. “This is not a protest,” she said, “this is chaos…. A protest has purpose…. If you love this city, if you care about this city, then go home.”

I’ve watched leaders give thousands of speeches, but I couldn’t get this one out of my mind. It was a remarkable and seamless blend of personal narrative, raw emotion and searing anger. And it was completely unscripted.

“I didn’t have a note, didn’t know what I was going to say,” Ms. Bottoms said this week in an interview. “When I was done, I thought, ‘either that went really right, or really wrong.’ ”

Most leaders in a crisis, whether they run a metropolis or a gastropub, prefer to work from a script. That’s doubly true for most CEOs, who issue careful statements or send emails to the staff. The problem with prepared statements is that any emotions they convey seem contrived. A great speech, on the other hand, fires the deep, unconscious emotional receptors in our lizard brains. It makes our hearts beat faster.

Most famous orations throughout history have been delivered in trying times, from the Civil War to the civil-rights movement. For the most part, however, they’ve shared a common theme: We will prevail.

What’s fascinating about the speech Ms. Bottoms gave is that she wasn’t trying to fortify people. She was imploring them to get their priorities straight.

It’s impossible to measure the impact of her words. Within two days, peace had largely been restored in Atlanta, although National Guard troops and a curfew surely helped.

Nevertheless, this speech offers something valuable to leaders in every field, especially in this challenging moment. It was a master class in handling one of the most delicate functions of management: constructive scolding.

Here’s a synopsis of the mayor’s techniques:

1. Ditch the Notes

Leadership 101 tells us that eye contact is vital and speakers shouldn’t spend too much time looking down at their notes. The less you look away, the stronger the spell.

Ms. Bottoms didn’t have any notes. She never broke eye contact.

2. Lose the Badge

When leaders deliver criticism, the first thing anyone sees is their authority. People process judgment from a leader differently than judgment from a peer. To show that they’re not just flaunting their power, leaders need to redefine themselves in the broadest possible human terms.

Ms. Bottoms began by saying: “I am a mother to four black children in America, one of whom is 18 years old. And when I saw the murder of George Floyd, I hurt like a mother would hurt.”

This brought her down to eye level with everyone.

3. Show Vulnerability

Many leaders believe that people won’t respect them if they admit to struggling with the same problems as everyone else.

Stoicism has its place, of course. In a video message sent to employees in the early days of the pandemic, Marriott CEO Arne Sorensen, who is undergoing cancer treatments, only briefly mentioned his illness. His goal, after all, was to lift spirits.

When leaders need to redirect people, however, vulnerability helps. It reduces the perceived distance.

Ms. Bottoms told her audience that when the violence started, “I called my son and I said, ‘Where are you?’ ” The fear showed on her face. Her voice quavered as if she might cry. “I said, ‘I cannot protect you. A black boy shouldn’t be out today.’ ”

4. Focus on Behavior

Nobody likes being judged. But when leaders attack a specific behavior, most people will at least consider their point of view. If you condemn their character, however, they’re more likely to tune you out.

In 2016, Facebook CEO Mark Zuckerberg learned that some employees were crossing out “Black Lives Matter” on office walls where staffers were encouraged to leave messages, and writing “All Lives Matter.”

In a sharply worded message, he wrote: “I was already very disappointed by this disrespectful behavior, but…I now consider this malicious as well.”

To a guilty party who might otherwise be persuaded, this statement crossed that invisible line. It left them with no clear path to redemption. It also suggested that the boss believes he’s morally superior.

Ms. Bottoms carefully framed her criticism of the violent protesters as a matter of perception, not virtue. “What I see happening on the streets of Atlanta is not Atlanta,” she said. Telling them, “you are disgracing our city,” was quite different from saying “You are a disgrace to our city.”

5. Be Fierce, But Offer Solutions

Next to fear, anger may be the most visceral human emotion. And after reeling her audience in, Ms. Bottoms let it rip. She conjured the fire of any loving parent who has truly had enough of this.

“You’re not protesting anything, running out with brown liquor in your hands, breaking windows in this city,” she shouted. “When you burn down this city, you’re burning down our community.”

Here, at the peak of volume, the mayor offered a better solution. “If you want change in America, go and register to vote,” she said. “Show up at the polls on June 9. Do it in November. That is the change we need in this country.”

6. Show Control

To perform well under pressure, psychologists say, people need to strike a balance between the instinctive, emotional and primal “red” functions of the brain and its logical, analytical “blue” functions.

In a crisis, most business leaders prefer to keep the “red” stuff under wraps. Last year, for instance, when Boeing CEO Dennis Muilenburg spoke publicly about the safety of the company’s 737 Max airplanes, he read from his notes at first, then took questions, without displaying any discernible emotion.

Ms. Bottoms made it clear, right away, that she hadn’t shown up in a lather. “Let me just speak to what’s happening here, today,” she calmly began. In the first minute, her pace was slow, controlled and deliberate.

Later, after fully unleashing her “red” emotion, the mayor quickly went “blue” again. She fell silent for three full seconds.

7. Reel It Back

Early on, Ms. Bottoms had leaned on two pronouns, “I” and then “you.” As her speech wound down, she added a third. “We are better than this,” she said. “We’re better than this as a city, we are better than this as a country.”

To reinforce this message of unity, she made an emotional shift. The vulnerability returned. She told the rioters to “go home” again, but this time it sounded less like an order than a mother’s plea. “In the same way I can’t protect my son…I cannot protect you out in those streets,” she said.

The demonstrations have continued in recent days with several hundred arrests. But the fuse has been extinguished. Ms. Bottoms visited protesters Thursday. “What I see right now as we’re all on our streets together; we are grieving together,” she said. “We are feeling each other’s pain. And the solutions will come.”

The mayor’s speech on May 29 seemed to simultaneously convey vulnerability and strength, passion and reason, shame and inspiration. Not everyone can pull that off.

If nothing else, however, any leader who needs to talk some sense into people can simply follow the mayor’s pronouns.

Start with “I,” pivot to “you” and finish with “we.”

—Mr. Walker, a former reporter and editor at The Wall Street Journal, is the author of “The Captain Class: A New Theory of Leadership” (Random House).

Write to Sam Walker at sam.walker@wsj.com