Klizan breezed to a 6-2, 6-2 first-round victory over Slovenian wild card Aljaz Bedene 6-2, 6-2 on the clay courts of Umag's ATP Stadium.
Kuznetsov, meanwhile, bested Spain's Daniel Munoz-De La Nava 6-3, 5-7, 7-6 (7-5).
Also on Day 1, Matthias Bachinger of Germany squeezed past Portugal's Joao Sousa 6-7 (5-7), 6-1, 7-6 (8-6).
Spaniard Fernando Verdasco and 2011 runner-up Marin Cilic of Croatia are the top seeds at this year's event.
Celebrity Families of 2014 [PHOTOS]