Lester (2-0) won his second World Series start, allowing four hits without a walk while striking out seven as the Red Sox notched a second straight win in St. Louis to grab a 3-2 series lead.
Koji Uehara recorded the final four outs for his second save of the series.
David Ross then lined a ground-rule double into the left-field corner, scoring Bogaerts with the go-ahead run.
After Lester grounded out, Jacoby Ellsbury blooped a run-scoring single to score Drew.
David Ortiz went 3-for-4 with an RBI, raising his World Series average to .733.
Wainwright (0-2) yielded three runs on eight hits over seven innings, walking one and striking out 10.