“This mandate was the right thing to do,” de Blasio said Monday. “We now see it worked.”

First responder and uniformed agencies have some of the lowest vaccination rates, though they have climbed in recent days: 77 percent of firefighters have received at least one shot, 83 percent of Sanitation workers, 85 percent of the NYPD and 88 percent of emergency medical services.

The mandate has led thousands of firefighters calling in sick, heated protests by city workers, unsuccessful lawsuits and garbage piling up in a slowdown by sanitation workers.

But de Blasio said agencies are filling in the gaps with overtime and extra shifts.

“We’re not seeing disruptions to any city services,” he said Monday. “People are picking up the slack.”

Since de Blasio announced the mandate on Oct. 20 for the city’s workforce of 380,000 people, 22,472 additional city workers have gotten the shot. Since a 5 p.m. deadline for employees to submit proof of vaccination, 3,564 people relented and got the jab before being turned away from work.

Teachers and public school staff and health care workers at the public hospital system were already required to get the shot under previous mandates. Among agencies covered by the latest order, the rate increased from 71 percent to 86 percent since the measure was announced.

On Monday, some 2,300 firefighters called in sick — and FDNY Commissioner Daniel Nigro said most of them are faking. Eighteen fire companies were out of service as a result, out of a total of 350 units. No firehouses were closed outright, but many companies were left short staffed.

“When a city employee fakes it and puts other people’s lives in danger, that’s a serious thing. And there’s going to be consequences for that,” de Blasio said, though he could not specify what disciplinary actions would be taken.

The mayor also warned that the city is scrutinizing union leaders who have protested the mandate for potential violations of labor laws. “We are watching every single thing they say and do — every email, every tweet, everything,” he said. “If we see a violation of the Taylor Law we will be in court immediately.”

The union representing firefighters on Monday called the mandate “unconstitutional” and complained of unvaccinated members being sent home, leading fire companies to close.

“This will put the lives of New Yorkers at risk and this is only the beginning,” the Uniformed Firefighters Association said in a tweet.

Suits against the mandate have thus far failed in state and federal courts.

Sanitation Commissioner Edward Grayson said he hopes to be caught up with garbage pickup by later Monday, after workers failed to pick up piles of trash bags in an apparent protest. They worked an extra shift on Sunday to clean up the mess.

“Once you get a little bit behind, it’s going to take a little while to right the ship,” he said. “We’re continuing to make up the backlogged service.”

At the NYPD, Commissioner Dermot Shea said there is “literally no effect on service” because most unvaccinated cops have applied for exemptions. Only 34 uniformed officers and 40 civilian employees have been placed on unpaid leave.

More employees may be barred from work in the coming days as decisions on exemptions are made — and de Blasio has said they will be granted very narrowly.

“I think you’ll see a number of people ultimately find that the exemption is not approved,” he said. “And then they still have that chance to correct, get vaccinated, come back.”

Workers on leave can return to the job at any time if they get the shot.

De Blasio urged other cities and states to impose a vaccine mandate similar to New York’s.

“To every mayor in America, every governor in America, to every CEO of a company in America — go to a full vaccination mandate,” he said. “It will allow us to end the Covid era once and for all.”