Croatia's Cilic had to go to three sets but moved on to the second round with a 6-3, 4-6, 6-4 victory over Ukrainian qualifier Sergiy Stakhovsky.
Meanwhile, Dolgopolov -- a Ukrainian who claimed his second career title this month in Washington -- was crushed by Russian former top-five star Nikolay Davydenko 6-1, 6-1 in a 52-minute match.
Also advancing Monday was former world No. 1 Lleyton Hewitt, who posted a 6-4, 6-4 victor over Russian Mikhail Youzhny.
Other Day 2 winners included American wild cards Brian Baker and Jesse Levine, Serb Viktor Troicki, Taiwanese qualifier Yen-Hsun Lu, Spaniard Pablo Andujar, Swiss Stanislas Wawrinka, Uzbek Denis Istomin, German Florian Mayer and Italian Andreas Seppi.