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New York City started to enforce its Covid vaccine mandate for municipal workers on Monday, and about 9,000 workers who refused to get vaccinated were placed on unpaid leave.

Mayor Bill de Blasio said on Monday that the vaccination rate for city workers continued to climb, reaching 91 percent, and that the city was running smoothly despite the absences.

“We’re not seeing disruptions to any city services,” he said, before urging workers to get vaccinated and return to work.

City officials said that they were coping with the reduction in workers without a significant slowdown of city services. With measures like shuffling firefighters between companies and extending the workday for sanitation workers, city agencies have been shoring up coverage.

Unvaccinated municipal employees without an approved medical or religious exemption — or a pending request for one — were placed on unpaid leave.

The number of vaccinated workers at city agencies has surged in the last few days, including at the New York Police Department, where 85 percent of employees were vaccinated on Monday, up from 70 percent when the mandate was announced.

Police Commissioner Dermot Shea said on Monday that placing unvaccinated officers on unpaid leave was having “literally no effect on service at this point.”

The city has a total of about 378,000 workers; about 6 percent remain unvaccinated, including those put on unpaid leave and about 12,000 workers who have applied for a medical or religious exemption. Those seeking exemptions are allowed to continue to work, while being tested weekly, until their cases are decided.

At the New York Police Department, officials had recently estimated that as many as 2,500 employees could be placed on leave on Monday as the department begins enforcing the vaccine mandate, according to a high-ranking official, who added that the number was falling because some personnel were getting immunized at the last minute.

The Police Department, which has about 36,000 uniformed officers and 19,000 civilian employees, was confident that it could manage the absences without taking measures such as canceling days off and moving to 12-hour shifts, said the official, who was not authorized to speak publicly.

The Sanitation Department has put workers on 12-hour shifts and told many to plan to come in on Sundays.

The vaccine mandate has been especially contentious within the tight-knit Fire Department. More than 2,000 New York City firefighters — out of a total uniformed force of about 11,000 — have taken sick days over the past week in what city officials describe as a large-scale protest against the mandate.

“Irresponsible bogus sick leave by some of our members is creating a danger for New Yorkers and their fellow firefighters,” the fire commissioner, Daniel A. Nigro, said in a statement. He attributed the uptick in sick leave to “anger at the vaccine mandate.”

The personnel shortage has put a strain on Fire Department operations. The department said that all its firehouses remained open, but that maintaining coverage across the city had required shuffling personnel around to reconstitute fire companies.

Andrew Ansbro, president of the Uniformed Firefighters Association, the union that represents rank-and-file firefighters, said that there had been no organized sickout. But hundreds of firefighters were feeling the side effects of vaccine doses and were too unwell to work, Mr. Ansbro said in a phone interview Sunday morning. “Hundreds of guys are feeling flulike symptoms, because that’s what the shot does to people,” Mr. Ansbro said.

The number of employees with approved exemptions — or pending ones — is unclear. But it could be that the number of people placed on unpaid leave on Monday is relatively small compared with the number that go on leave in the days ahead — should those applications for exemptions be mostly rejected.

The Police Department, for instance, has received applications for exemptions for some 6,500 officers, according to the official. Over the next two weeks, the department is expected to decide most of those cases, the official said, adding that a large portion of the requests would most likely be denied.