Second seed Djokovic continued his bid for a record-equalling 24th major title with a 6-2 7-5 6-4 win in New York.
“I did well in the important moments and I’m pleased to get through in three sets,” said three-time winner Djokovic.
The 36-year-old Serb will play ninth seed Taylor Fritz, who is one of three American men in the quarter-finals.
Fritz, 25, won 7-6 (7-2) 6-4 6-4 against Swiss qualifier Dominic Stricker, joining Frances Tiafoe and Ben Shelton in progressing on Sunday.
Tiafoe, 25, beat Australia’s Rinky Hijikata 6-4 6-1 6-4 to set up a last-eight meeting with Shelton, guaranteeing at least one home player in the semi-finals.
Shelton, 20, beat 14th seed Tommy Paul 6-4 6-3 4-6 6-4 as three American men reached the quarter-finals for the first time since 2005.
Fritz, Tiafoe and Shelton are each bidding to become the first man from the host nation to win the singles title since Andy Roddick in 2003.
In order for Fritz to emulate Roddick, he will have to find a way past Djokovic first.
Djokovic reached a 57th major quarter-final, leaving him one adrift of Roger Federer’s record in the Open era, with a routine victory over 105th-ranked Gojo.
The manner of the win was the polar opposite to the fightback from two sets down which he had to perform against fellow Serb Laslo Djere in the previous round.
Djokovic did not concede a break point to US Open debutant Gojo in the opening set, but lost his serve for 2-0 in the second by planting a volley into the net.
However, 25-year-old Gojo was unable to consolidate. The big-serving Croat hit five double faults as he handed over the game without making Djokovic apply any real pressure.
Djokovic continued to nullify Gojo and smartly took care of his own service games, breaking three more times as Gojo became increasingly weary and took a third match point with an ace out wide.
“I knew Borna has a big game and is a big-shot player. One of the keys to the match was to reduce his serve and stop the aces,” said Djokovic.
“I tried to manage my energy and focus on accuracy with my serve.
“The matches are obviously going to get tougher but I’m ready.”
