The Yankees opened Tuesday night having lost four in a row and in last place in the AL East with an offense that was in the dumps.

So when they fell behind by two runs with Gerrit Cole on the mound, it seemed things might get even worse.

But they tied it in the sixth— with the help of an Anthony Volpe homer — and took the lead for good thanks to an unlikely source, Willie Calhoun, in a 4-2 victory over Cleveland in The Bronx.

Calhoun, signed to a minor league deal in the offseason and part of what’s been a wildly unproductive outfield group, hit a solo shot to right with one out in the seventh to make it 3-2.

The win got the Yankees back above .500 (16-15) and was just their third victory in their past 10 games.

After the Yankees’ bullpen did them in during Monday’s loss, Ron Marinaccio, Wandy Peralta and Michael King pitched three scoreless innings in relief Tuesday.

Though Cole was far from his best, he gave up just two runs over six innings as the Yankees improved to 7-0 in his starts this year. And he again avoided giving up any home runs. He’s yet to give up any after allowing a career-high 33 a year ago.

Cole issued leadoff walks in both the first and second innings. He pitched around the free pass to Steven Kwan in the first and Josh Bell in the second.

After walking Bell, Cole hit Andres Gimenez with a pitch then fell behind Mike Zunino, 1-0. That prompted Jose Trevino to head to the mound for a chat with Cole, who responded by striking out Zunino, Will Brennan and Myles Straw — all looking.

Cole couldn’t avoid trouble in the third, when he gave up a pair of runs. The rally started with back-to-back hard-hit singles by Amed Rosario and Jose Ramirez with one out. Josh Naylor then smacked an RBI double to right for the game’s first run.

Cole then walked Bell to load the bases and force a visit from pitching coach Matt Blake. Cole responded by getting Gimenez on a soft grounder to first that scored Ramirez to make it 2-0 before Zunino struck out looking again.

On offense, the Yankees got just a two-out single from Anthony Rizzo in the first off right-hander Tanner Bibee — in just his second major league start — until Volpe hit an opposite-field homer with one out in the bottom of the sixth to make it 2-1.

Gleyber Torres followed with a double to right to knock Bibee out of the game.

Rizzo tied the game with an RBI single off Bibee’s replacement, James Karinchak, and Calhoun gave the Yankees their first lead with one out in the seventh, taking Karinchak deep to right on an 0-2 pitch for his first home run of the season.

“We expect these moments,’’ Aaron Boone said before the game of the recent adversity. “You’ve got to deal with them and grind your way through them. We’re equipped to handle that. We know there’s gonna be stretches of the season when it’s not real easy, whether it’s injuries or [we’re] scuffling, it’s part of it. We’re confident as a group we’re gonna be a good team.”