The defending champion struggled through the first set in hot and humid conditions in New York but won it on a tie-break and looked much better from there on.
Murray dominated the second and third sets to run out a 7-6 (7/2) 6-2 6-2 winner in one minute short of two hours.
The Scot said: "It was very tough conditions. He's a very tricky opponent. He plays strange shots and he takes your time away, it's tough to get into a rhythm. I'm really glad to come through in three sets.
"It cooled down a little bit towards the end but the first and second sets it was very hot. We've been told a thunderstorm is coming so hopefully that will take away some of the humidity."
Having beaten Leonardo Mayer not entirely convincingly on Friday, Murray faced a tricky customer in the Argentinian, ranked 47. A tricky customer with an unorthodox double-handed slice backhand, he had given Murray a tough time in both their previous meetings.
The Scot would have been happy to be back on Arthur Ashe after a trip to Louis Armstrong, where he always seems to struggle, and he was given a great welcome even though the stands were far from full.
The first game contained plenty of drama, with three break points, two HawkEye challenges and two overrules, but Mayer eventually held.
Murray did not look very comfortable, holding his left hip in the way he does when his troublesome back is playing up.
At times his first serves were well under 100mph, and his first-serve percentage was very low, but he was still not in any real danger of being broken.
And he made a great start to the tie-break, winning the first four points, which proved more than enough of a lead to clinch the set.
Murray had loosened up now and he took an early lead in the second set with a break in the second game, drilling two backhand winners down the line.
He had to save a first break point in the next game, coming up with a big first serve, but he was playing much closer to the baseline and with a lot more energy.
His groundstrokes were flatter and a fizzing forehand brought up a first set point on the Mayer serve, which Murray took.
The conditions looked to be sapping Murray's energy so finishing the match in three sets was a priority.
He took a big step towards that with another break in the fourth game of the third set, dismissively smacking a forehand return winner past the German.
Mayer did not look like he had any belief he could beat the world number three, but he staved off more break points to hold for 4-2.
Murray just wanted to get over the finish line and he did not have to wait too long. A terrific forehand dink at full stretch helped him set up two match points and he took the first when Mayer netted a forehand.
Murray, who will play either Andreas Seppi or Denis Istomin next, added: "I need to start matches a little bit quicker, I was a little bit slow out of the blocks."