The marathon victory allowed the Red Sox to jump back into first place in the American League East, a half game in front of Tampa Bay.
Seattle's Lucas Luetge (0-2) was sent out to work his third inning in the 15th and he opened the frame with a four-pitch walk to Pedroia. David Ortiz grounded out to first, sending Pedroia to second, and Mike Napoli was walked intentionally.
After Jarod Saltalamacchia struck out, Jonny Gomes drew a walk that loaded the bases. That brought up Drew, who finally ended the game with his base hit to right.
The contest took 5 hours, 3 minutes to play and brought the Red Sox their fourth victory in five outings.
Seattle missed scoring chances after putting two runners on base in the 11th, 12th and 15th innings.
Drake Britton (1-0) worked the final two innings to collect his first major league win in his sixth relief appearance.