On the day Albert Pujols left the St. Louis Cardinals in December 2011, their general manager, John Mozeliak, slipped out of the winter meetings unnoticed and bolted for the airport. There were four hours before his flight was due to leave Dallas, but Mozeliak needed to go somewhere to compose himself. He found solace in an American Airlines lounge. "I was down, depressed, disheartened," he said.

Eighteen months later, Mozeliak and the Cardinals are hardly ruing the day Pujols, the iconic slugger, signed a 10-year,...