Murray blitzed Lukas Rosol of the Czech Republic 6-3, 6-2, 6-2. The reigning Wimbledon champion and Olympic men's singles gold medalist never faced a break point.
"(I had) good variation on the second serve," Murray said. "Once I got the break, I started to relax and played well."
At the opposite end of the victory-difficulty scale was 19th-seeded American Steve Johnson, who staved off six match points to edge Russia's Evgeny Donskoy 4-6, 1-6, 7-6 (2), 6-3, 6-3.
Johnson saved three match points after falling behind 2-5, 0-40 in the third set, then three more while trailing 5-6.
"I didn't look so good for a while," Johnson said. "At 2-5, I just found a way to hold. I got lucky and then we were back even. I have no idea how I got out of that game and just found a way to win those points. My mentality was that it's not over. My goal walking out today was to win in three sets, but it doesn't matter how it comes.
"After I finally broke, I started to play better and better. Then I won the third and I'm thinking that I'm in good control of the match even though I was down two sets. It doesn't always come easy, and it would be boring if it was that way."
Japan's Kei Nishikori, who previously experienced some struggles making it out of the first round at the U.S. Open, didn't encounter any such issues Tuesday. He posted a 6-1, 6-1, 3-6, 6-3 victory over Germany's Benjamin Becker.
The sixth-seeded Nishikori had been ousted in the first round in three of the past five years. He reached the final in 2014 when he didn't exit early.
Nishikori was definitely on his game against the 35-year-old Becker.
"He started playing much better so it was tough third and fourth sets but luckily I got out of the last two games," Nishikori said.
Nishikori faces Russian Karen Khachanov in the second round. Khachanov defeated Italy's Thomas Fabbiano, 6-3, 6-4, 4-6, 6-3.
Wawrinka fought off Verdasco's charge in the opening set and finished off the sweep.
The loss was just the second for Verdasco in 14 career first-round matches at the U.S. Open.
Twelfth-seeded David Goffin of Belgium fell 4-6, 7-5, 6-4, 6-0 to American qualifier Jared Donaldson. Australia's Bernard Tomic, the 17th seed, also was an upset victim, falling 6-4, 6-3, 4-6, 7-6 (0) to Damir Dzumhur of Bosnia and Herzegovina.