Russian Kudryavtseva is ranked 98 places below her opponent but that did not stop her claiming the first set with ease before the Croatian fought back to claim the second.
The third set went to a tie break but Tomljanovic ran out of steam as Kudryavtseva completed a 6-1 3-6 7-6 (7/2) win.
Daria Gavrilova was another Russian to make it into the second round as she beat compatriot Vera Dushevina 7-6 (7/2) 7-6 (7/3) but Olga Puchkova was defeated 6-1 6-3 by Belgian Alison Van Uytvanck.
German Dinah Pfizenmaier beat Canada's Stephanie Dubois 6-7 (7/9) 6-3 6-4 in the day's remaining first-round match.
There was more Russian success in the final qualifying round with Ekaterina Bychkova beating Pole Paula Kania 3-6 6-4 7-5 and Valeria Solovyeva claiming a 6-0 4-6 6-3 win over Ukraine's Olga Savchuk.
Chin-Wei Chan gave the home fans something to cheer about with a 6-4 6-1 win over Tereza Mrdeza of Croatia while Lesley Kerkhove won her all-Dutch battle against Arantxa Rus 6-4 6-4.