The seventh seed was dealt a blow in the first set when Lepchenko took the edge by grabbing the first break point of the set.
After losing the break, Jankovic was able to hold her service games, but the damage was already done as the American took the lead.
Jankovic had a task on her hands in the second set, but she regained composure by sealing three break points and keeping her opponent on the back foot in order to draw level.
The players could not be separated in the third, taking the set to a tie-break, and it was the world number 49 who came out victorious, stunning her opponent with a string of powerful winners to secure a place in the next round.
Lepchenko will face Croatia's Ajla Tomljanovic in the next round.