7:29 PM ET

NEW YORK — Houston, here come the Yankees.

Riding off a strong start from lefty Nestor Cortes, the New York Yankees booked a ticket to the American League Championship Series with a 5-1 victory over the Cleveland Guardians on Tuesday. The Game 5 victory capped a back-and-forth series in which the Yankees faced elimination in Game 4 in Cleveland and rain delayed multiple contests.

“We’ve had talks in that clubhouse that we’re family, no matter how bad things go or how good things can go, we’ve got to stay level-headed,” Cortes said. “We’ve got ourselves, we don’t have anybody else to lean on.”

They will need more of the same in ALCS facing off against the Astros in a best-of-seven series that starts in Houston on Wednesday when Justin Verlander and Jameson Taillon go head-to-head in Game 1. During the regular season, the Astros and Yankees finished with the two best records in the American League, respectively, while Houston took five of seven games against New York.

“The challenge is we’re facing a great team,” said Yankees manager Aaron Boone. “As far as the quick turnaround, we’ll be fine. That’s baseball. We do that all the time. We’ll walk in there with some confidence. We know they are a great team and rested and ready.”

To match up against Houston, the Yankees will need the offense that showed up on Tuesday night against the Guardians. Giancarlo Stanton put the first runs on the scoreboard in the first inning Tuesday, hitting a line-drive home run on a 87.5 mph cutter from Cleveland starter Aaron Civale over the right field wall, scoring Gleyber Torres and Anthony Rizzo and putting the Yankees on top 3-0. Civale lasted one more batter, throwing just 26 pitches and recording a single out before Guardians manager Terry Francona gave him the hook.

The early run set the tone for Cortes on the mound.

“I knew from that moment on, all I had to do was throw strikes and be able to get us as deep as possible,” Cortes said.

Aaron Judge tagged on another run in the second inning, hitting a 81.3 mph curveball from Sam Hentges over the right-center field wall to make it 4-0. It was Judge’s fourth homer in a winner-take-all game — the most in major league history — and the 13th homer for the towering slugger in his postseason career, catapulting him past Reggie Jackson and Yogi Berra for the fifth-most in Yankees postseason history.

“We’re just trying to do our job,” Judge said. “When you’re in the middle of the order, you gotta come up and produce. Stanton and Rizzo have been doing it all series and I tried to step in there and do what I can.”

The Yankees faced a scary moment in the third inning when shortstop Oswaldo Cabrera collided with outfielder Aaron Hicks on a pop-up to left field off the bat of Cleveland outfielder Steven Kwan. Hicks left the game with a left knee injury and underwent an MRI at New York Presbyterian. He will miss the remainder of the postseason.

Cleveland struck back with its only run of the game later in the third inning when third baseman Jose Ramirez hit a sacrifice fly to center fielder Harrison Bader, scoring Austin Hedges to make the score 4-1. The run proved to be the only offense the Guardians generated in their final game of the season, tallying eight hits while leaving six runners on base and going 0-for-6 with runners in scoring position.

“We pushed them about as far as we could and they came out and beat us today,” said Guardians manager Terry Francona. “They get to move on and we don’t.”

Rizzo tagged on another run for the Yankees in the fifth inning when he knocked an RBI single to center field off Guardians reliever James Karinchak to drive in Torres, who stole second base to move into scoring position after drawing a walk.

“We controlled the zone,” Boone said. “That was when we are at our best, making you come into the zone. … It was just heavy at-bats and competitive at-bats really set the tone for us.”

Cortes left the game after the fifth inning after throwing 61 pitches, allowing just three hits, one run and one walk while striking out two batters. He was working off of just four days of rest, last pitching on Friday in Game 2. Throughout the outing, Cortes used a mix of his fastball (54%), cutter (26%) and slider (16%), taking the approach of trying to induce weak contact versus trying to strike out batters.

While Cortes found out on short notice that he would be starting the series finale, hearing from Boone immediately after the game was rained out on Monday night, the Yankees dugout agreed that Cortes had better stuff on Tuesday than he did in his previous ALDS start.

“Just the legend of Nestor,” Boone said. “I knew he wanted the ball, and I just felt like once we got rained out, I felt convicted that Nestor was who I wanted to go with.”

Cortes leaned into the pressure that came with starting an elimination game.

“I’ve got nothing to lose,” Cortes said. “I’ve been in the gutter before. I’ve been down, so for me, this is just a cherry on top and hopefully there’s more moments like this. I feel like today, I came in with the responsibility in my hands to take care of the narrative, to take care of what my career has been. I was just happy to perform.”

The departure of Cortes after the fifth handed the keys to the Yankees’ fate to the bullpen, which appeared to be the team’s biggest weak spot heading into the postseason. First, Boone called upon Jonathan Loaisiga, who allowed singles to Amed Rosario and Ramirez before getting Oscar Gonzalez and Josh Naylor to ground out before striking out Gabriel Arias to end the sixth inning. Loaisiga returned for another inning and finished off the seventh, ending the evening with two innings pitched, allowing three hits and no runs or walks while striking out two.

Clay Holmes added to the optimism around the Yankees’ bullpen, entering in the eighth inning and shutting down the top of the Guardians’ order, striking out Rosario and Gonzalez and inducing a groundout from Ramirez.

“I feel like [Holmes] has been throwing the ball really well,” Boone said. “Locked in. He’s come in some big spots, and I feel like back a month or so before he had the shoulder, I feel like he started to really get it going. I thought he was sharp in the series.”

Holmes then handed the ball off to Wandy Peralta, who’s become one of Boone’s go-to relievers during the series, pitching in every game against Cleveland. Peralta got Naylor to ground out before allowing a single to Arias, striking out Andres Gimenez and allowing a single to Luke Maile. With two runners on, Peralta got Myles Straw to ground into a fielder’s choice and punched New York’s ticket for a matchup against the Astros.

The Yankees did not have much time to spray champagne and drink beer, with the team bus leaving for the airport leaving about an hour-and-a-half after the last out. Had Game 5 not been rained out on Monday, the Yankees expected a more festive plane trip to Houston. But with no off day before kicking off the ALCS, the team expects to be mostly business.

“I’m hoping we do some scouting reports on the plane so we can sleep in,” said Yankees pitcher Gerrit Cole.

Multiple Yankees admitted that if they had an off day before Game 1 of the ALCS, they would likely be drinking more beer and champagne. Instead, the team ended the festivities quickly, wrapping up in around 20 minutes before packing up their stuff to head to the airport.

“We’ve got other plans,” Judge said. “We’re definitely gonna celebrate and enjoy the moment, but we have a big series coming up starting tomorrow.”

The Yankees and Astros face off in Houston in Game 1 of the ALCS on Wednesday starting at 7:37 p.m. ET.