Reliever Sam Tuivailala was nearing the end of his second inning of work in extras before he gave up the go-ahead run.
With two outs in the top of the 13th, Mitch Moreland hit a ground-rule double just out of the reach of left fielder Tommy Pham. After manager Mike Matheny intentionally walked Jackie Bradley Jr., pinch hitter Chris Young laced a single to score Moreland from second.
The run gave the Red Sox their first lead of the night at 5-4.
The Cardinals have Jedd Gyorko, Yadier Molina and Aledmys Diaz due up in the 13th.