The highest seed in the draw following Angelique Kerber's withdrawal, Svitolina found things far from straightforward as she required three sets to see off wildcard Heather Watson 6-2 5-7 6-3.
Watson let her serve fall to love to hand the Ukrainian the opening set, responding well by forcing a decider with a well-timed break in game 11 of the second.
Svitolina upped her game, though, proving far more efficient to complete the win despite letting two match points slip through her grasp beforehand.
Strycova defeated Yulia Putintseva 6-3 6-3 in just one hour and 23 minutes, perhaps leaving time for dinner with a fan after refusing to rule out the option in her on-court interview.
Japanese teenager Naomi Osaka edged out Lauren Davis 6-1 2-6 7-6 (7-4) and Naomi Broady beat Alize Cornet 7-6 (7-3) 6-0 in the remainder of the action.