Nunez, back in the Boston's lineup Thursday after missing all but one of the final 20 regular-season games, had to be carried off the field in the first inning of Game 1 of the ALDS against the Houston Astros after aggravating a sprained ligament in his right knee.
The Red Sox then made the roster request to MLB, which approved it Thursday night.
Nunez will be eligible to return in the World Series.
Young, the former longtime Arizona Diamondbacks outfielder, batted .235 with seven home runs and 25 RBIs this year in his second season with the Red Sox.
Nunez stumbled out of the batter's box after bouncing a first-pitch grounder to third base at Minute Maid Park. He began limping down the line, then fell to the dirt before reaching first base. The starting designated hitter was batting second for the Red Sox, who lost 8-2 to the Astros.
"We did a lot of treatment, a lot of simulated games, we did everything we can to be ready," Nunez said. "We obviously know now my knee's not ready yet."
Nunez will undergo further imaging before the Red Sox decide whether to remove him from the postseason roster.
The Red Sox were hoping Nunez would be ready to play even though he already had suffered one setback in his recovery from a sprained posterior cruciate ligament. His previous attempt to get back in the lineup lasted for two at-bats last week before he had to be lifted from a game against the Toronto Blue Jays.
Nunez was fitted for a brace, which the Red Sox hoped would help stabilize his knee when he twisted during his swings and began running out of the batter's box. After a workout Monday at Fenway Park, the Red Sox were confident Nunez would be able to return in the postseason.
Manager John Farrell said the Red Sox have "no regrets" about trying to bring Nunez back for the division series.
"He had been cleared prior to the start of this series," Farrell said. "The number of tests we put him through, we felt like he had satisfied all that we could simulate. But it was apparent trying to go game speed -- we saw what transpired."
Acquired from the San Francisco Giants a few days before the trade deadline, Nunez gave the Boston offense a jolt. He batted .321 with eight homers in 165 at-bats after hitting only four homers in 302 at-bats with the Giants.
Nunez, known for his versatility, settled in as the every-day second baseman in August while Dustin Pedroia was on the disabled list.
Hanley Ramirez replaced Nunez in Thursday's game and had two hits.
Information from ESPN's Scott Lauber was used in this report.