The Serbian's chances of repeating her 2012 success at the WTA International event in Azerbaijan appeared slim after a disastrous opening set, but she rallied to win 2-6 7-5 6-3.
Mayr-Achleitner made a sensational start to the match winning five games in a row as Jovanovski only collected seven points, and the Austrian sealed the opener having converted her fifth set point.
Jovanovski showed great battling spirit in the second, upping her first-serve percentage to 76, and she levelled the match with three successive games having broken her opponent three times.
Momentum in the deciding set swung between both players before Jovanovski stamped her authority on proceedings at 2-3 with a strong finish, this time taking four consecutive games to seal progression.
She will meet Katarzyna Piter next after the Pole also came from a set down to win her opening match against Kristyna Pliskova 6-7 (4-7) 6-3 6-4.
Last year's beaten finalist Shahar Peer got her campaign under way with a comfortable win against sixth seed Yvonne Meusburger.
The Israeli fell to defeat at the final hurdle 12 months ago to Elina Svitolina but showed the hard-court event suits her style with a 6-2 6-4 win on Monday.
Stefanie Vogele and Pauline Parmentier also had no problems in the first round with straight-set wins, while Urszula Radwanska advanced at the expense of Alison Van Uytvanck, who retired with injury after losing the opening set 6-1.