Nathan Eovaldi dominated the reigning World Series champs from the hill, allowing just two runs and six hits in six innings of work. The Boston bullpen allowed only one hit over the final three innings of the triumph at Minute Maid Park.
Boston now leads the series 2-1.
The Red Sox center fielder stepped up to the plate with two outs in the frame before providing the dramatic long ball. Bradley had a 1-1 count with the bases loaded as the Red Sox held a 4-2 advantage.
He smashed Roberto Osuna's third offering of the exchange over the right center field fence for a 386-foot grand slam. The ball had an exit velocity of 101-mph, according to Statcast.
Boston built its initial lead on an RBI double from J.D. Martinez and an RBI groundout by Xander Bogaerts in the top of the first inning.
Marwin Gonzalez cut the Red Sox's lead in half with an RBI single in the bottom of the inning for the Astros.
Houston tied the game in the fifth frame on an RBI double from Alex Bregman. But Red Sox first baseman Steve Pearce gave Boston the lead once again with a solo shot to left field off of Astros reliever Joe Smith, making the score 3-2 in the sixth inning.
Mitch Moreland was hit by a pitch with the bases loaded for the Red Sox's fourth run in the top of the eighth inning. Bradley came to the plate next to give Boston some insurance.
Game 4 of the 2018 ALCS is set for 8:39 p.m. on Wednesday in Houston.