Murray won 6-2 6-3 to set up a third-round clash with Julien Benneteau, who beat Radek Stepanek 6-2 5-7 7-6(4).
“I wanted to play a solid match, but I would have liked to have hit a few more winners. I came to the net a few times and missed a couple of volleys, but I was getting into positions,” Murray told Sky.
“I want to play more aggressively [before the US Open].”
The Scot, playing in only his second tournament since winning Wimbledon, raced into a 4-0 lead at the start of the match with two breaks of serve and his Russian opponent could not find a way back into the set from there, failing to convert his one break chance.
The second seed again broke twice to earn a 4-1 lead in the second stanza, and although Youzhny broke one game back on his sole break point in the set, Murray was comfortable in closing out the win in an hour and 10 minutes.
Youzhny will be unhappy with the seven double faults he served – against one from Murray – while the twice Grand Slam champion dominated both on first- and second-serve.
Murray lost in the third round of the Masters tournament in Montreal last week to Ernests Gulbis after a month’s break from the court, and reached the final of the doubles.
“It was an important match for me to get through,” Murray added. “I want to get through as many matches as possible before the US Open. I only had a couple last week.
“I prefer the conditions here. The court suits my game. I felt more comfortable and moved pretty well.”
The final Grand Slam of the year at Flushing Meadows begins on August 26.