Lou Piniella appeared headed to another loss against Pedro Martinez, who pitched seven innings and left with a 4-1 lead. Ramiro Mendoza breezed through the eighth for Boston, but Alan Embree failed to retire any of the three hitters he faced in the ninth.
Travis Lee led off with a single and Shumpert, signed Saturday after he was released by the Los Angeles Dodgers, lined a 1-0 pitch over the left field fence to make it 4-3.
After Ben Grieve singled, Red Sox Manager Grady Little summoned Chad Fox (0-1), who struck out Damian Rolls, retired Brent Abernathy on a forceout and threw two quick strikes to Crawford. But the veteran of 64 major league games fouled off four pitches and took a ball before depositing Fox's next offering in the right field seats.
It was Crawford's third career home run.
The rally made a winner of Seth McClung (1-0), who struck out three and walked two in two scoreless innings in his major league debut.
It was a disheartening loss for Boston, which allowed Ugueth Urbina to leave via free agency and opted to rotate a handful of veterans in the closer's role.
2014: The Year in Music [PHOTOS]