This Sunday, the Tony Awards will honor the best of Broadway and likely deliver many of the biggest prizes to “Hamilton,” the Public Theater’s biggest hit since “A Chorus Line” had its debut in 1975.

So when the nonprofit theater had its annual gala Monday night in Central Park, it seemed not just like its biggest fund-raiser to date (the company raised a record $2.5 million), but a kind of pre-Tonys celebration and victory tour for its artistic director, Oskar Eustis.

Over the last four years, he has presided over one action-packed season after another, with successes that included “Here Lies Love” (David Byrne and Fatboy Slim’s musical about Imelda Marcos, wife of the Philippine dictator Ferdinand E. Marcos), “Fun Home” (Lisa Kron and Jeanine Tesori’s musical about a young lesbian whose father commits suicide) and Danai Gurira’s “Eclipsed” (a harrowing drama about genocide in Liberia that is currently nominated for six Tony Awards).

“We’ve had a good couple of years,” Mr. Eustis said earlier that day, shortly before guests like Bette Midler, David Remnick (the editor in chief of The New Yorker) and Julianna Margulies streamed into Central Park to start another season of Shakespeare in the Park.

Mr. Eustis didn’t have a quick answer for what explains this extraordinary run. It’s partly trial and error, he said, the way he has learned to develop new musicals, which was something he didn’t really know when he took the job in 2005. He has also watched his operating budget increase to $36 million last year, from $21 million in 2008.

But he has come to believe that success builds upon success, just as connections build upon connections. And that dynamic was on ample display Monday night as Mr. Eustis stood by the Delacorte Theater.

One moment, he was chatting with Michael R. Bloomberg, who had arrived with his girlfriend, Diana Taylor. The next, Mr. Eustis was huddled on the lawn with Samantha Power, the United States ambassador to the United Nations, who is a major aficionado of the Public Theater, having taken colleagues and diplomats to “Hamilton,” “Eclipsed,” “Fun Home” and Shakespeare in the Park.

“Oskar is my soft power New York projectile,” Ms. Power said. “If you want to change the world, the Public is the place to take people. Their whole perspective changes in ways that it doesn’t trying to argue the case with them rationally.”

In this increasingly plutocratic era, there was a retro quality to the way Mr. Eustis bounced around the room in his seersucker suit. Even at 58, more than 20 years after he became an accomplished director, known partly for his work on the original production of “Angels in America,” Mr. Eustis retains a strangely boyish quality.

With his thick mop of brown reddish hair and his bow tie nearly upside down, he looks almost like a cross between a Dr. Seuss character and the fifth member of the Talking Heads.

When the pop singer Sophie B. Hawkins came over to Mr. Eustis after the grilled chicken dinner, and told him that she was writing a musical, he barely seemed to know what to do. “I’m actually a little verklempt,” he said, the minute she left. “I had such a crush on her when I was young.”

Nevertheless, Mr. Eustis has also been around long enough to have the sort of wisdom no man wishes for.

When the financial crisis struck in 2008, donations dropped off precipitously. An attempt to bring the musical “Bloody Bloody Andrew Jackson” to Broadway in 2010 turned into a sinkhole that led to serious unrest on the board. In 2014, Mr. Eustis’s teenage son died, a tragedy that Mr. Eustis described simply as “a horror.”

At last year’s gala, Mr. Eustis was honored and nearly broke down in tears as he spoke about what the experience had been like for his entire family, which includes his wife, Laurie, and his daughter, Kyle Brown. So as Mr. Eustis posed for pictures with Lin-Manuel Miranda on Monday night, and squeezed the wrists of board members like Alexandra Shiva, the contrast to years past was impossible to miss.

And it was nice to see the mood remained festive as attendees moved into the theater around 8 p.m., where Mr. Eustis ascended to a lectern at the front of the stage. There, he thanked his staff, the board chairwoman Arielle Tepper Madover and his sponsors at Bank of America.

He also gave a little shoutout to the talent in the room.

“Shakespeare’s a really great writer, but just sitting in Section C we have David Hare and Lin-Manuel Miranda,” he said, segueing into his introduction of the evening’s program: a tribute to Shakespeare’s influence on America.

Over the next two hours, actors like Bill Irwin, Lily Rabe, Andre Holland and Hamish Linklater performed scenes from “Othello,” “Hamlet” and “Romeo & Juliet,” while Jimmy Smits, F. Murray Abraham and Phylicia Rashad read the writings of historical figures like John Quincy Adams, Abraham Lincoln and John Wilkes Booth, discussing how Shakespeare had influenced them.

Then, just before 10 p.m., the show’s director, Jeremy McCarter, came up to the stage with James Shapiro, the editor of the recent book “Shakespeare in America,” to introduce the final performance: Meryl Streep and Christine Baranski. Ms. Streep and Ms. Baranski were there to do the duet “Brush Up Your Shakespeare,” from “Kiss Me Kate,” Cole Porter’s 1948 musical update of “The Taming of the Shrew.”

And, as Mr. McCarter explained, both parts ordinarily played by men would be played by women.

It soon became clear what he meant. Ms. Streep’s skin had a certain orange hue. Her hair was sprayed blond and went about a half-foot high. And she had on a black suit and a red tie, like the ones Donald J. Trump wears on the campaign trail.

She was even doing Mr. Trump’s trademark gesticulations and facial expressions: pursing her lips, rolling her eyes and shrugging with disdain, as she sang about being rejected by women.

Half the audience seemed to be gasping for air, they were laughing so hard. Ushers ran down the aisles to tell patrons that video recording was verboten. It was an oxymoron if ever there was one: a viral theater moment.

“That was one for the ages,” said Ms. Midler, shortly after it ended and cast members took their bows.

Had Mr. Eustis known what was about to go down?

“I had an idea,” he said a little while later, sitting on a bench with his wife as dessert was served and a D.J. played for the crowd.

In fact, Mr. Eustis continued, there had been a certain amount of discussion among his staff members that mocking the presumptive Republican nominee for president may not be such a smart idea.

But deciding whether to encourage Ms. Streep was among his easier executive dilemmas. “She’s Meryl Streep,” he said, flashing a smile.