The seventh seed had lost her last two matches on the surface and started badly at the WTA International event on Monday, as Van Uytvanch took the opening five games on her way to winning the first set.
However, Jovanovski came roaring back and recorded a 1-6 6-4 7-5 victory in just over two hours.
Having wasted her first two match points, the 22-year-old eventually secured the win with her sixth break of serve.
Third seed Klara Koukalova also had to come from a set down to book her second-round place as Donna Vekic's spirited start fizzled out.
In a match strewn with errors from both players, it was the 32-year-old who prevailed with a 4-6 6-0 6-4 win.
Vekic edged the opening set, but her game fell apart in the second as Koukalova dominated.
After winning six consecutive games to level the match, Koukalova extended that run to 10 and stormed into a 4-0 lead in the decider.
Vekic was able to mount a comeback and level, but an eighth loss of serve saw the Czech progress after 95 minutes on court.
The third match on Monday saw fourth seed Elina Svitolina beat Danka Kovnic 6-1 4-6 6-4.