It was Kuzmova who enjoyed an early nerve-settler as she out-rallied her opponent to win her first service game.
Viktoria Kuzmova impressed during a tight opening set
Image credit: Getty Images
But the number sixth seed - who had Gael Monfils supporting in her box - created momentum in the first set with quick holds of service, and Kuzmova had to save three break points in the sixth to keep the opening set on serve.
Having wriggled out of a tough spot, the Slovakian showed more quality ball-striking, a backhand opening up the court before a forehand down the line produced another strong service hold to make it 4-4.
Monfils supports Svitolina from player's box
Svitolina then saved three break points before a timely ace allowed her to move within one game of winning the first set.
Kuzmova was given no short balls to step up the court, and Svitolina took her first set point when her opponent hit a backhand into the net.
Svitolina raised her game further and continued to give very few cheap points away as she broke once more to take control of the second set in the fourth game.
Highlights: Svitolina swats aside Kužmová in straight sets
Another forehand winner at 40-15 allowed the 2018 WTA Finals champion to back up her break before she pulled clear as her opponent ran out of ideas.
"I was trying to be light on my feet and I think this was the key today," Svitolina said afterwards. "Winning the event in Singapore was something that brought me a lot of confidence and I've just tried to transfer this to the Grand Slam."
Svitolina speechless after question over relationship with Monfils