BOSTON (CBS) — The Boston Red Sox are moving on. And the New York Yankees are going home.

In the must-win, do-or-die Wild Card playoff game at Fenway Park on Tuesday night, the Red Sox were the better team from start to finish, winning 6-2 in front of a sold-out crowd.

Xander Bogaerts and Kyle Schwarber homered for Boston, while Alex Verdugo drove in three runs.

Nathan Eovaldi made it through five shutout innings before surrendering a solo home run in the top of the sixth inning. He struck out eight batters while allowing the one run and four hits with zero walks in his 5.1 innings of work.

Gerrit Cole lasted just 2-plus inning, failing to record an out in the bottom of the third inning before being lifted. He allowed three runs, all of which came on homers, while giving up four hits and two walks and striking out three Red Sox.

The Red Sox got out to a 2-0 lead in the bottom of the first inning after Rafael Devers worked a two-out walk. Bogaerts then launched a 427-foot home run to straightaway center field.

The lead stretched to 3-0 for Boston in the bottom of the third inning, when Schwarber turned on a high fastball and sent a no-doubter over the visiting bullpen in right field.

The Yankees finally got on the board in the top of the sixth, when Anthony Rizzo jumped on a curveball from Eovaldi and curled it around the Pesky Pole for a solo homer.

Following Rizzo’s homer, Eovaldi induced weak contact from Aaron Judge, who hustled out his chopper to short for an infield single. Alex Cora then brought out the hook for Eovaldi — a surprising move, considering Eovaldi had retired 11 straight Yankees before the home run and looked unhittable for most of the night.

The move almost backfired immediately, too, as Ryan Brasier allowed a line drive from Giancarlo Stanton to deep left-center field, high off the Monster. Brasier and the Red Sox were fortunate, though, as Yankees third base coach Phil Nevin made the ill-advised decision to send Judge home. The relay from Kiké Hernandez to Bogaerts to Kevin Plawecki arrived at home with plenty of time to spare, allowing the Red Sox to record the second out of the inning at the plate.

Brasier then induced a pop-up from Joey Gallo to end the inning and the threat.

In the bottom of the sixth, Bogaerts worked a one-out walk, and he scored from first base on a double to deep  right-center field by Verdugo.

Verdugo did the honors of blowing the game open one inning later, when he stepped to the plate with the bases loaded and two outs following three Red Sox walks. Facing Chad Green, Verdugo sent a line drive over the shortstop’s head, driving home two runs, before getting thrown out at second base to end the inning. The Red Sox led 6-1 at that point, with Verdugo driving in half of the team’s runs.

Hansel Robles pitched a 1-2-3 eighth inning, and Garrett Whitlock entered to pitch the ninth, allowing a one-out home run to Stanton before getting the final outs of the game.

Red Sox pitchers walked zero Yankees in the game, while the Yankees issued seven walks.

Game 1 of the ALDS will take place Thursday night in Tampa.