All-star shortstop Jhonny Peralta figured he would attract plenty of offers as a free agent in 2013, but when the St. Louis Cardinals expressed interest, Peralta told his agent to stop shopping him around—just close a deal with the Cards as soon as possible.

Like most players, Peralta wanted the best chance to win, and the Cardinals afforded it. They begin their division series Friday, against the Chicago Cubs, with baseball’s best record. They are in the playoffs for the 12th time in the past 16 years.

It...