NEW YORK — To extend their season another day, the New York Yankees needed that big, momentum-shifting hit that eluded them over the first three games of the World Series. They got it on the brink of elimination Tuesday night in Game 4 at Yankee Stadium.

Anthony Volpe belted a go-ahead grand slam in the fourth inning, Austin Wells added a solo shot two innings later, and Gleyber Torres put the finishing touches on an 11-4 win over the Los Angeles Dodgers with a three-run home run in the eighth inning as the Yankees avoided elimination for at least one night.

Volpe’s breakthrough blast came on a first-pitch slider from Daniel Hudson, the second reliever the Dodgers deployed for their scheduled bullpen day, with two outs and the Dodgers leading 2-1. The grand slam, which landed a few rows beyond the wall in left field, electrified a sellout crowd that had been on edge after Freddie Freeman’s two-run home run in the first inning.

It was Volpe’s first career grand slam and the first by a Yankee since Tino Martinez in Game 1 of the 1998 World Series against the San Diego Padres. Volpe, at 23 years and 184 days old, became the youngest Yankee with a grand slam in the World Series since Mickey Mantle in 1953.

It was not a promising start for the Yankees. Freeman’s home run — a laser over the short porch in right field off Yankees starter Luis Gil that extended his home run streak in the World Series to a record six games — immediately deflated the crowd. It put the Yankees, who hadn’t led since Freeman’s walk-off grand slam in Game 1, in an early hole for the third straight game. But the Dodgers’ reliever carousel could not hold the Yankees down.

Five Yankees relievers, meanwhile, were not charged with a run over the game’s final five innings. The quintet of Tim Hill, Clay Holmes, Mark Leiter Jr., Luke Weaver and Tim Mayza held the Dodgers without a hit after the fifth inning. It was the formula the Yankees had been seeking in this series. It saved their season Tuesday.