Having enjoyed a quickfire win over eighth-seeded Spaniard Marcel Granollers in his opening encounter, Tomic had a bigger fight on his hands on Wednesday with Kavcic pushing him to three sets before the Australian prevailed 6-3, 4-6, 6-4.
Tomic's father John, who was banned from gaining accreditation to ATP events for 12 months, was again spotted in the stands with tournament organisers allowing him to attend as a spectator.
Tomic came bursting out of the blocks to race to a 4-1 lead before wrapping up the first set.
But the Australian seemed to lose his momentum after a brief rain delay was called in the second set.
And up 5-4 when they returned Kavcic took advantage, breaking Tomic's serve right away to level at a set each.
Ahead 5-4 in the final set, Tomic closed out the match on his second match-point with an overhead smash.
The world No. 52 Australian will now take on either Ukrainian Alexandr Dolgopolov or second-seeded Pole Jerzy Janowicz for a spot in the semi-finals.