From the general manager’s box at the Metropolitan Opera, Peter Gelb surveyed the hall that has been his domain for nearly 20 years. It was 10 minutes before curtain one evening in April, the orchestra was warming up, and there was just enough time to check the hall for empty seats — very few that night — and to steer his guests, including the director of a new opera about Frida Kahlo and Diego Rivera, to their places in his box.

This was the final performance of “Innocence,” the searing work by the Finnish composer Kaija Saariaho about a school shooting. “Innocence” was one of a number of Met productions this season that won the acclaim of audiences and, to a certain extent, critics — “Tristan und Isolde,” “I Puritani,” “La Sonnambula” and “The Adventures of Kavalier & Clay” among them.

But for all the success onstage, Gelb is under siege, struggling to steer the Met through a fiscal crisis that is imperiling its position as the most ambitious opera company in the country. Gelb is an old-school New York character trying to ride the generational, cultural and financial changes that are sweeping not only the Metropolitan Opera but also opera houses around the world.

In the latest setback, the Met faces a $30 million shortfall after Saudi Arabia pulled out of a collaboration Gelb had negotiated; it was expected to provide $200 million to the company over the next eight years. This blow came after the Met tapped nearly one-third of its endowment to cover operating expenses.

But as he finishes his 19th season, confronting doubts about the viability of the Met as well as key parts of his strategy to rescue it, Gelb conveys confidence that he will again find a way to save the opera house that he said he always dreamed he would run. He is resilient and relentless, a former record company executive and agent who, in his own telling, has rarely met a challenge he couldn’t beat.