BOSTON—As the sun set over Fenway Park on Saturday evening, Mariano Rivera saved a Yankees victory over the Red Sox for the 57th time in his career. A couple of hours later, he sat with his wife and children at a corner table at Abe and Louie's, a famous local steakhouse. And when he stood up to leave, something remarkable happened: People started to clap.

The applause started with a few tables. Then a few more. And by the time Rivera was done posing for photos and chatting with complete strangers, everyone in the restaurant...