6-4 7-5 to reach the second round of the Masters 1000 event.
U.S. teenager Jared Donaldson is threatening to join them after winning the first set 6-2 against world number four Stan Wawrinka.
The 18-year-old wildcard twice broke the second seed in winning the opening set, helped by a rash of unforced errors from Wawrinka, before the Swiss settled down and swept the first three games of the second set.
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Play was then called for the day with the return of more rain.
Rain through the day and into the night plagued the hardcourt run-up to the U.S. Open, leaving four men's matches and three women's contests still to finish to complete the first round.
Double Olympic gold medallist Andy Murray is waiting to face the winner of a match between Juan Monaco of Argentina and Ivo Karlovic of Croatia. Monaco leads 6-4 5-3.
American Johnson told Tennis Channel it had been a stop-start day.
"I can't keep track of how many times we stopped and started and warmed up," he added. "But I'm super happy to come out and get the win."
Johnson was hoping to build on his run in Rio where he paired with Jack Sock to medal in doubles, and took eventual gold medallist Murray of Britain to a third-set tiebreaker in the singles quarter-finals.
Another U.S. teenager tasted defeat as Russian Mikhail Youzhny dismissed 18-year-old Taylor Fritz 6-1 6-3.
The Russian qualifier advanced to a second-round match against Japan's Kei Nishikori.
Gael Monfils of France, seeded ninth, beat Spain's Pablo Carreno Busta 6-3 6-4, while his 13th-seeded compatriot Richard Gasquet battled past another Frenchman, Adrian Mannarino 7-6(2) 3-6 6-1.
Isner said he felt he might be poised to improve on a "mediocre" season.
"I had a very good serving rhythm. I'm happy with how I served," the world number 22 said. "I feel like I have begun to turn the corner a little bit."
Next up for Isner is fourth-seeded Canadian Milos Raonic, the world number six, who has lost to the American in all three of their previous meetings.
(Reporting by Larry Fine in New York; Editing by Peter Rutherford)
