In a Fed Cup tie last month, Sprem defeated Russian Elena Likhovtseva, but lost to Anastasia Myskina.
"This is a big day for me and by far my best win," Sprem said. "The best result I had before that match was taking Myskina to three sets in Fed Cup. Coming into the match, I didn't believe I could win today even if my coach always tells me that I am the best and that I should believe in myself.
"I guess I'll believe him after that match."
Dokic lost in the final here last year to Italian Sylvia Farina Elia and reached the quarterfinals at the French Open. The 20-year-old Dokic will enter the second Grand Slam of the year next week having lost three straight matches on clay.
"What can I say? I played a very bad match today, it was definitely not my day," said Dokic, who will be seeded seventh at the French Open. "I made so many double faults throughout the match and she took advantage of it. I played better in the second set, but made too many unforced errors in the third."
Myskina, the second seed here, avoided an upset as she dispatched American Corina Morariu, 6-3, 7-6 (7-4). The 21-year-old Myskina is seeking her third title of the year.
Farina Elia, the seventh seed, dispatched German Anca Barna, 6-3, 7-5, and sixth-seeded Russian Vera Zvonareva had an even easier time, routing Tina Pisnik, 6-1, 6-0.
American teenager Ashley Harkleroad also moved into the quarterfinals. Harkleroad held a 6-3, 5-0 lead when Russian eighth seed Elena Likhovtseva retired. Harkleroad will next meet Frenchwoman Emilie Loit, who overcame Switzerland's Marie-Gaianeh Mikaelian, 4-6, 6-4, 6-4.
Earlier in the day, Loit completed her first-round match against third-seeded Russian Elena Dementieva, 6-3, 2-6, 6-3.
French teen-ager Marion Bartoli also needed three sets to beat Russian Lina Krasnoroutskaya, 7-6 (7-3), 3-6, 7-5.
Maja Matevzic of Slovakia earned a quarterfinal meeting with Sprem by cruising past American Laura Granville, 6-3, 6-4.
2014: The Year in Music [PHOTOS]