Defending champion Djokovic was 4-2 ahead in the first set when Russian 34-year-old Youzhny called it quits suffering from a left hamstring injury.
The outcome completed a bizarre first week for world number one Djokovic. After labouring over four sets to beat Jerzy Janowicz on Monday, he was handed a walkover into the third round when Czech opponent Jiri Vesely withdrew from their second round clash with an arm injury.
“I have never had this happen before in my Grand Slam career, getting a walkover and the next match lasting just half an hour,” said Djokovic.
The top seed next faces Britain’s world number 84 Kyle Edmund who stunned John Isner of the United States 6-4, 3-6, 6-2, 7-6 (7/5) to make a Slam fourth round for the first time. Nadal reached the fourth round for the first time since 2013 — the year of his last New York title — with a 6-1, 6-4, 6-2 win over Russia’s Andrey Kuznetsov.
“Andrey is always a tough opponent, he returns well and has good shots from the baseline,” said fourth seed Nadal who faces France’s Lucas Pouille for a quarter-final slot.
“I had a fantastic first set, a little trouble with my serve in the second but the third was key so I’m very happy.” Djokovic’s very brief encounter had been preceded on Arthur Ashe Stadium by women’s eighth seed Madison Keys pulling off an epic comeback to beat Japan’s Naomi Osaka.
Keys won the latest-ever finishing women’s match at the tournament on Monday when she completed victory over compatriot Alison Riske at 1:48 am.
On Friday, the 21-year-old was involved in more drama when she rallied from 1-5 down in the final set to defeat Osaka 7-5, 4-6, 7-6 (7/3). The 18-year-old Osaka was so traumatised by her failure to convert her dominance into victory that she broke down in tears as she twice tried and failed to serve for the victory.
Jack Sock, the American 26th seed, reached the fourth round for the first time by knocking out 2014 champion Marin Cilic 6-4, 6-3, 6-3 in tie where he did not face a single break point.
The 23-year-old Sock, whose last two appearances at the US Open had ended in retirements, goes on to face French ninth seed Jo-Wilfried Tsonga.