Former world number one Jankovic burst into a 3-0 lead in the first set after securing an early break, but Cirstea did respond at 0-3 to move onto the board.
The respite did not last long, however, with Jankovic securing two more breaks, the second of which allowed the Serbian to move one set ahead in the contest.
It was Cirstea that secured the first break of the second and what proved to be final set, but the Romanian's own serving continued to suffer.
Jankovic broke back at 2-1 to level before securing back-to-back breaks in the latter stages, with the 29-year-old taking her second match point at 2-5.
Jankovic will face Sara Errani in the next round.