The British number one secured his spot in the second round with a 7-5 3-6 6-2 win over the 10th seed.
It was their third meeting of the year and Evans moved 2-1 up in the score count after two hours and 32 minutes of action in New York.
It took the world number 28 three breaks before he could secure the opening set, but it was the Russian who came out of the blocks quickest in the next.
After Rublev had levelled, the decider immediately went the way of Evans who established a 3-0 lead before he closed out to set up a second-round meeting with Milos Raonic, who reached the final of Wimbledon in 2016.
"I am happy with how I played in the third, it was really deserved and some good tennis as well," Evans told Amazon Prime.
"It will probably be a little less physical match (against Roanic) which is nice, but I spent plenty of time out here so it will be difficult.
"Everyone knows how good he is and I will be looking to put my game on the court and try to beat him."
Evans was the second British player to reach the second round in New York after Andy Murray defeated Frances Tiafoe on Saturday.
The duo could meet in the last 16 if two-time Wimbledon winner Murray can stun fifth seed Alexander Zverev on Monday.