The Astros sent Game 1 starter Justin Verlander to the mound, who easily cruised through the first two innings.
But the Red Sox struck first in the third inning on a home run to left field by J.D. Martinez, giving Boston the early 1-0 lead.
Hopes of a comeback continued to unravel in the sixth inning.
After hits from Mitch Moreland and Ian Kinsler, Red Sox outfielder Rafael Devers hit a 3-run homer to extend Boston's advantage to 4-0.
Houston eventually got on the board in the seventh inning after a solo home run by Marwin Gonzalez.
But at the end, Houston couldn't get any baserunners in, and the Boston Red Sox eliminate the Astros with a 4-1 final.