Argentina Davis Cup star Schwartzman, is seeded 24 in Melbourne, and he secured his place in the last 16 of the first grand slam event of the year with a superb performance against the Ukrainian.
Dolgopolov took the first set on a tie-break 7-1, but Schwartzman showed great character to win the next set 6-2.
He stepped up a gear to win sets three and four 6-3 in a contest lasting just over two and a half hours. The victor served 13 aces, including a fastest of 171kmh.
Schwartzman, 25, looks set to play former winner Rafa Nadal who led was well on the way to beating Damir Dzumhur after winning the first two sets.