Before the match, British number two Norrie said he was “ready for anything” from seventh seed Djokovic – including seeing his opponent struggle with fitness issues before regaining his level.
However, the world number 35 was still not able to drag Djokovic into gruelling physical exchanges as he hoped.
Norrie briefly threatened to cause a shock when he moved a break up early in the third set.
But Djokovic instantly hit back – helped by his opponent hitting a double fault on break point – before regaining control with a precise serving game which Norrie largely failed to trouble.
Djokovic became the oldest man to reach US Open fourth round since Jimmy Connors, who was also aged 38, in 1991.
“I’m still trying to find my groove. Today I played the best I have so far in the tournament,” Djokovic said.
On his fitness, he added: “You have some ups and downs but you don’t want to reveal too much to your rivals. I’m as young and as strong as ever.”
