Both players stumbled by losing serve in their opening games of the first set, but it was Makarova who broke again to give herself the edge in the encounter.
Svitolina tried to stop her opponent's set point with two chances of a break, but Makarova held firm to take the lead on the Arthur Ashe Stadium.
The momentum was firmly with Makarova as she broke serve at the first time of asking in the second set, and while Svitolina tried to respond by creating three break points, she was unable to take them.
Eventually, the 20-year-old managed to break to make it more of a contest with games going by way of serve, but Makarova edged ahead on break points to successfully serve for the match, despite suffering cramp in her leg.
Makarova will face Kristina Mladenovic in the next round.