It was the second time since 2011 that five Yankees players homered in one game.
"It was nice to see," manager Joe Girardi said. "Our team is built for this park."
Brett Gardner and Alfonso Soriano also homered off a struggling Edinson Volquez to back five scoreless innings from David Phelps (1-0). Brian McCann added a two-run drive in the eighth against reliever Vin Mazzaro as New York won for only the second time in eight home games.
Coming off consecutive road shutouts against the crosstown rival Mets, the Yankees took their third straight following a four-game slide.
"We had a very good game overall," Soriano said. "That's what we expect."
Starling Marte homered and had three hits in his return to the starting lineup for the Pirates, winless at Yankee Stadium since Harvey Haddix pitched them to a 5-2 victory in Game 5 of the 1960 World Series.
Volquez (1-4) gave up six hits -- four homers -- and no walks in 6⅓ innings, dropping to 0-4 in his past five starts.
Teixeira connected on a 1-2 pitch following Derek Jeter's single in the first for his eighth home run in his past 17 games. It was the 350th of his career, tying Chili Davis for sixth place among switch-hitters.
Almonte sent Volquez's first pitch of the third into the raised bleachers in right-center for his second major league homer. Called up from the minors Tuesday, Almonte was a late addition to the lineup when center fielder Jacoby Ellsbury was scratched with flu-like symptoms.
Gardner homered leading off the sixth and Soriano did the same in the seventh.
"They are great hitters and I made a lot of mistakes," Volquez said. "You cannot make a lot of mistakes in this ballpark."
After a rainout Friday night, 47,353 fans turned out on a beautiful afternoon for the Pirates' first visit to the current Yankee Stadium. They went 0-6 in interleague games at the old place, swept in a June 2005 series and again two years later.
Phelps pitched out of early trouble to earn his first win as a starter since July 4, 2013, at Minnesota. After beginning the season in the bullpen, the right-hander was making his third start since replacing injured Michael Pineda in the rotation.
"At the end of the day it was five zeros and I'll take that," he said. "You don't want to grind every time out, but it's one thing that I've kind of done this year. I haven't really had my best stuff yet."
Phelps got some help from the overaggressive Pirates, who ran into three outs on the bases.
"We were pushing guys out there," manager Clint Hurdle said. "Those are the chances you want to take when you're not getting those big hits with runners in scoring position, try to push yourself across the plate or get yourself in scoring position."
Phelps is 3-0 with a 1.08 ERA in his past three starts at Yankee Stadium.
"He made pitches when he had to," Girardi said. "There were some good plays."
Pirates bopper Pedro Alvarez, who grew up a few miles from Yankee Stadium, went 0-for-4 with two strikeouts in his first game against the Yankees. He left about a dozen tickets for family and friends and was expecting his father to be in the crowd.
Two calls were overturned on replay challenges, one in favor of each team. Both plays involved tags by Jeter at second base.
"We won with the home run today, which we're capable of doing," Teixeira said. "We have to score a little bit more and I think the last few games our pitching has really picked us up. So it's good to see us break out today."
Marte (lower back tightness) made his first start in a week. ... 1B Ike Davis, traded by the Mets last month, went 0-for-3 with a walk and three strikeouts in his return to New York. ... Girardi said he would be "kind of shocked" if LHP CC Sabathia (right knee) is able to return after only 15 days on the disabled list. "We don't know when he's coming back," Girardi said. ... Pineda (shoulder muscle) is scheduled to throw 25 pitches on the side Sunday. ... Ichiro Suzuki (back) entered in RF in the ninth and is likely to play in the traditional doubleheader Sunday -- the first single-admission twinbill at Yankee Stadium since 2004. ... RHP Hiroki Kuroda chose to start the opener Sunday against Pirates RHP Charlie Morton. LHP Vidal Nuno will pitch the second game against RHP Gerrit Cole, who spurned the Yankees to attend UCLA after they drafted him out of high school.