Shiv Puri gathered his employees together after the lunchtime rush last week to deliver the big news: This year, the cook and kitchen staff at the Bombay Sandwich Company on West 27th Street will receive paid sick leave for the very first time.

“It’s a full week,” Mr. Puri explained as he handed out a fact sheet describing New York City’s first paid sick leave law. “That includes whether you’re caring for yourself or a member of your family.”

His eight employees were thrilled. No surprise there. But Mr. Puri, a new business owner, was feeling pretty good, too, after weeks of worrying and calculating the cost of the new law.

“It’s the law and it’s the right thing to do,” he said. “It won’t bust the bank. It won’t put us in jeopardy.”

The law went into effect on April 1. And despite the naysayers and the critics, the skies didn’t fall. Instead, without hoopla or hullabaloo, the city quietly became the largest in the nation to ensure that a vast majority of workers wouldn’t lose their jobs or a portion of their paychecks if they or their close relatives got sick.

For Mayor Bill de Blasio, who significantly expanded the law’s reach when he took office in January, it is a notable and welcome victory. He has been battered and bruised...