NEW YORK

Just a couple of days into the season’s second half, Yankees manager Joe Girardi said his team needed to win this one.

Fortunately for New York, he had the right men on the mound.

Masahiro Tanaka out-pitched David Price in their first matchup to win his fourth straight decision, and the Yankees ended Boston’s six-game winning streak Sunday night with a 3-1 victory over the Red Sox.

“I think everybody understood how important it was to win today,” Tanaka said through a translator.

Starlin Castro keyed New York’s three-run fourth inning with a double that scored Didi Gregorius from first base, snapping a 10-game drought without an RBI.

No. 9 batter Austin Romine lined a go-ahead single with two outs off the glove of diving second baseman Dustin Pedroia. Jacoby Ellsbury added another RBI single off Price, who gave up 11 hits in 5 2/3 innings as the Yankees prevented a three-game sweep.

“We had to bounce back,” Gregorius said. “For us, today showed that we’re fighting.”

Following a loss on Saturday, Girardi said the Yankees had to turn things around immediately and called Sunday night “probably as important (a) game that we’ve had in July in a long time.”

New York responded with excellent pitching and a handful of timely hits.

“When you write up a win for us, this is exactly what we need,” Romine said.

All-Star slugger Carlos Beltran even called it “a must-win” game. Well aware he could be an attractive trade target if the Yankees (45-46) decide to sell before the Aug. 1 deadline, Beltran said the current 10-game homestand is important to “what the future holds for a lot of guys here.”

Price (9-7) had only one strikeout following three consecutive outings with 10. Pedroia homered early for Boston, which remained two games behind first-place Baltimore in the AL East.

“We’re right in this all the way through the end,” manager John Farrell said.

Tanaka (7-2) fanned seven with one walk over six innings of three-hit ball in his latest stingy performance on extra rest. Pitching with six days between starts, the Japanese right-hander improved to 6-0 in 11 starts with more than four days off.

He is 1-2 with a 5.33 ERA in his other eight games.

“More rest means more time to sort of get the inflammation out of your body,” Tanaka said.

Tanaka had Red Sox hitters in baseball’s highest-scoring lineup chasing sliders and splitters below the strike zone. He threw just 87 pitches — but that was all the Yankees needed because their three big bullpen arms were all fresh and ready to go.

“I thought he gave us everything he had and I thought he aired it out in the sixth inning,” Girardi said. “We have those three big guys down there for a reason, so I’m going to go to them.”

Dellin Betances, Andrew Miller and Aroldis Chapman combined for three hitless innings. Chapman worked the ninth for his 18th save in 19 chances, and the Yankees upped their record to 17-1 when all three pitch in the same game.

New York’s bullpen has thrown 19 straight shutout innings.

Boston hasn’t swept a series at Yankee Stadium since June 2011, and the Red Sox haven’t won seven in a row overall since June 1, 2014.

“You’re the guy who goes out there and doesn’t help us win that seventh one, that’s tough,” Price said.

Notes — Price, who leads the AL with 141 strikeouts, fell to 14-9 in 34 career games (33 starts) against the Yankees — including 1-2 with a 7.79 ERA during his first season in Boston. The left-hander gave up six hits to left-handed hitters (in 11 at-bats), matching a career high.

Red Sox — RHP Junichi Tazawa (shoulder impingement) threw all his stuff in a 40-pitch bullpen session. But the team would like Tazawa to face hitters in a simulated game — likely Tuesday — before activating him from the 15-day disabled list, Farrell said.

Up next for Red Sox — Following a day off, RHP Rick Porcello (11-2, 3.66 ERA) starts Tuesday night at Fenway Park against former Boston RHP Jake Peavy (5-7, 5.09) and the major league-leading San Francisco Giants in the opener of a nine-game homestand. That begins a stretch of 43 scheduled games in 44 days for the Red Sox.