The 10th-seeded Isner converted 5 of 9 break points while saving 6 of the 7 he faced.
In addition to Isner, Americans Sam Querrey and Steve Johnson also advanced Monday. Unlike his countrymen, Jack Sock was ousted in the first round, falling 6-2, 7-6 (12), 1-6, 5-7, 6-4 to Australian Jordan Thompson.
The fifth-seeded Cilic was playing his first match since losing to Roger Federer in the Wimbledon final last month. No man has ever won the US Open after not entering any tournaments between the All England Club and Flushing Meadows.
Cilic didn't even practice for two weeks because of a left leg injury. But he produced 55 winners Monday, 33 more than Sandgren.
The 105th-ranked Sandgren, whose name comes from his Swedish great-grandfather and has nothing to do with tennis, pumped his fist after he held serve to win the third set.
Ferrer made a surprise run to the semifinals of the hard-court Cincinnati Masters earlier this month.
In other matches, 12th-seeded Pablo Carreno Busta defeated American Evan King 6-3, 6-2, 7-6 (5); Edmund of Great Britain defeated No. 32 Robin Haase of the Netherlands 6-3, 7-5, 6-3; and British qualifier Cameron Norrie defeated Russian Dmitry Tursunov, who retired after dropping the first two sets 7-6 (7), 6-1.
Zverev, who upset Andy Murray on his way to a quarterfinal run at this year's Australian Open, got all he could handle from the 710th-ranked Virginia Cavalier, whose collegiate title in May earned him a wild card into the US Open main draw.
Also advancing Monday was Denis Shapovalov, who put away Russia's Daniil Medvedev, 7-5, 6-1, 6-2. The 18-year-old Canadian became the youngest player to reach the semifinals of an ATP Masters 1000 event at the Rogers Cup in Montreal earlier this month, but had to qualify to get into the US Open main draw. The 69th-ranked Shapovalov will meet eighth-seeded Frenchman Jo-Wilfried Tsonga, a 6-3, 6-3, 6-4 winner over Marius Copil of Romania.
Information from The Associated Press was used in this report.