CLEVELAND — The Yankees’ most established pitcher delivered exactly what the Yankees needed on Saturday — and one of their emerging relievers did the same for the first time.

After Gerrit Cole shut down the Indians for most of seven innings — outdueling Shane Bieber — Jonathan Loaisiga closed it out with a four-out save, and the Yankees won their third straight game for the first time this season.

The 2-1 victory at Progressive Field came after manager Aaron Boone said he wanted Cole to “set the tone and keep this train going.’’

Cole allowed just a fourth-inning run while giving up just three hits, no walks and striking out 11. Bieber, last year’s AL Cy Young winner, allowed a pair of fifth-inning homers to Aaron Hicks and Rougned Odor — and the pitching took care of the rest.

“I felt it was what we needed,’’ Hicks said of his homer. “It was a pitchers’ duel until then, and it changed the momentum to our favor.”

The same could be said of their trip to Cleveland, where the Yankees have begun to put the miserable start to their season behind them.

“It’s nice to get it turned in the right direction,’’ Hicks said.

That’s due as much to Loaisiga as anyone else.

Justin Wilson took over for Cole in the bottom of the eighth and walked Roberto Perez on four pitches. The lefty then fell behind pinch hitter Jordan Luplow 3-0 — while also firing a wild pitch to send Perez to second. But Wilson got Luplow to pop out.

Boone said afterward he wasn’t sure who would close with Aroldis Chapman out after finishing the previous two games.

Loaisiga came on to face Amed Rosario and got the former Met to pop to short to end the inning before coming back in the ninth to face the heart of Cleveland’s lineup: Cesar Hernandez, Jose Ramirez and Eddie Rosario, and retired the side in order for his first major league save.

“It definitely felt different,’’ Loaisiga said through an interpreter. “You realize you’re going into the game to face the best hitters and understand where the game is. … When you warm up out there, you definitely feel a little pressure. After I threw the first pitch, all that disappeared.”

“I’m glad he capped it off,’’ Cole said.

“We’re definitely starting to play better,’’ Boone said of his team, which has won four of five since losing five in a row. “They’re building on their confidence and starting to walk in here expecting things to happen.’’

The Yankees had a chance to get to Bieber in the top of the first. Giancarlo Stanton singled up the middle and Aaron Judge walked with one out, but Gleyber Torres struck out and Gio Urshela stared at a called third strike.

Bieber went on to retire 11 consecutive Yankees following Judge’s walk.

Cole retired the first seven batters he faced until Perez drilled a 3-2 four-seamer for a double to right-center with one out in the third.

Cleveland broke through in the fourth. Ramirez hit a fly ball to the wall in right, where a leaping Judge got his glove on it to save a homer. But Judge couldn’t hang on and Ramirez hustled to third for a one-out triple. Rosario followed by serving a single into left to give Cleveland a 1-0 lead.

It didn’t last, as Hicks led off the fifth with a homer from the left side, where he was 4-for-41 entering the at-bat.

“Huge,’’ Cole said of Hicks’ homer. “Huge. There’s no other word to describe it. And the cherry on top was Odor.”

Following strikeouts by Clint Frazier and Kyle Higashioka, Odor homered for a second straight game to give the Yankees a 2-1 lead with a 426-foot shot to right-center.

Cole finished his night by retiring the final 11 batters he faced, leaving after seven innings and a season-high 111 pitches. Cole has struck out 42 in his past 26 ¹/₃ innings — and walked just one.

“It was two special guys going against each other,’’ Boone said of Cole vs. Bieber. “[Cole] answered the bell again.”