Josh Donaldson celebrates with teammates after hitting a walk-off RBI single in the 10th inning Thursday night against the Boston Red Sox. The Yankees won, 5-4. Frank Franklin II/Associated Press

NEW YORK — Aaron Judge fell a few feet short of a record-tying 61st homer, hitting a 404-foot drive caught in front of the center-field wall in the ninth inning, but the New York Yankees beat the Boston Red Sox 5-4 Thursday night on Josh Donaldson’s 10th-inning single to clinch their sixth straight playoff berth.

Judge had walked three times and struck out once before he came to the plate with the score tied 4-all in the ninth. The crowd of 43,123 was on its feet for every pitch, and Judge drove a 2-2 fastball from Matt Barnes just to the right of straightaway center.

The ball left the bat at 113 mph, and fans waited in anticipation as Judge jogged toward first base. But they groaned in unison as Kiké Hernández made the catch a step in front of the fence, not far from the 408-foot sign.

Judge also threw out a runner at second base to help hold off Boston in the ninth, showing off his defensive prowess. Tommy Pham hit a one-hopper off the right-field wall against Clay Holmes (7-4). Judge played the carom, and from the warning track threw a strike to shortstop Isiah Kiner-Falefa for the tag. It was Judge’s sixth assist of the season.

Then in the 10th, with pinch runner Marwin Gonzalez at second as the automatic runner, Kaleb Ort (0-2) intentionally walked Gleyber Torres before Donaldson grounded a single just past diving shortstop Xander Bogaerts and into left field.

Giancarlo Stanton hit a two-run homer for the Yankees, and Harrison Bader had a tying sacrifice fly in the eighth after Stanton’s leadoff infield single.

New York is headed to the playoffs for the 24th time in 28 years, and the Yankees have a 7 1/2-game lead in the AL East.

Triston Casas hit a solo homer and pinch-hitter Reese McGuire delivered a three-run drive in a four-run seventh off Clarke Schmidt as Boston overcame a 3-0 deficit.

CHOPPY CHAPMAN

Aroldis Chapman had another shaky outing for the Yankees in relief, walking two batters with one out in the eighth – the second on four pitchers.

TRAINER’S ROOM

Red Sox: Second baseman Trevor Story (bruised left heel) went back on the 10-day injured list and 1B/3B Bobby Dalbec was recalled from Triple-A Worcester.

Yankees: Left-hander Zack Britton (Tommy John surgery) was activated from the 60-day IL and left-hander Wandy Peralta (back) was put on the 15-day IL, retroactive to Monday.