The world number nine had to dig deep in his first meeting with the Russian despite dominating the first set, before ultimately coming through 6-1 7-5 4-6 6-2 to seal his progress.
Murray had already beaten Robin Haase and Matthias Bachinger in straight sets to reach the third round at Flushing Meadows for a ninth straight year, and he seemed to be cruising to another straightforward victory after taking the opener.
The 2012 champion did so for the loss of just one game as he began in positive fashion, although he was forced to save two break points in the third game.
Kuznetsov was more of a threat in the second - helped mainly by errors that started to creep into Murray's game - but the Scot still seemed to be on course when an ace sealed him a two-set advantage.
To his credit, though, the Russian continued to battle.
With his serve suddenly starting to find a way through, he forced Murray onto the defensive - the seed also continuing to contribute to his own problems as he produced three double faults, one when facing set point, and failed to take three of the four break-point opportunities he carved out.
Before he knew it, Murray was 4-1 down and although he hauled it back to 4-4, a wild forehand and a volley missed their target, before he double faulted when serving to stay in the set.
Kuznetsov perhaps sensed an opportunity, but any hope of a fightback was quickly snuffed out in the fourth.
Murray had suffered cramp in his opening match against Haase on the same court but there were no concerns this time as he stepped up his game when it mattered most.
There was one final nervy moment when his serve faltered at 4-1 up, but it proved only a minor stumbling block as he wrapped up the set 6-2 to book a meeting with either Jo-Wilfried Tsonga or Pablo Carreno Busta.
Afterwards, Murray insisted he was making steady progress as the tournament enters its second week.
"I just tried to stay solid. It was a tough match but I feel better than I did in the first match," he said.
"I made sure I did everything properly. I love it here. I have enjoyed coming here since I was a junior.
"I play some of my best tennis here."