The 2014 champion won 6-4 6-3 3-6 6-3 in two hours and 47 minutes.
Spain’s 21st seed David Ferrer lost 6-4 3-6 2-6 1-6 to Mikhail Kukushkin.
American 13th seed Jack Sock was another early casualty, losing 6-2 7-6 (14-12) 1-6 5-7 6-4 to Australian Jordan Thompson in just under four hours on Louis Armstrong Stadium.
French eighth seed Jo-Wilfried Tsonga beat Romanian Marius Copil 6-3 6-3 6-4, while American 10th seed John Isner won 6-1 6-3 4-6 6-3 against France’s Pierre-Hugues Herbert.
Tsonga will play Denis Shapovalov in the second round, after the Canadian beat Russia’s Daniil Medvedev 7-5 6-1 6-2, while Isner faces South Korea’s Chung Hyeon.
Sandgren, ranked 105th in the world, had been due to play Andy Murray,before the Briton’s withdrawal on Saturday.
“I feel just a little bit rusty,” said fifth seed Cilic after Monday’s win.
“I need a few matches to get in the rhythm. I was hitting some good shots and then making some unforced errors,” added the world number seven, who will play German Florian Mayer in the second round.
American Sam Querrey, who reached the Wimbledon semi-finals, beat Frenchman Gilles Simon 6-4 6-3 6-4, while Luxembourg’s Gilles Muller was a 3-6 6-3 6-4 6-4 winner over Australian Bernard Tomic.