Swiss Bencic, 18, was playing for the first time as a member of the elite after achieving the No 9 ranking with last weekend's St Petersburg final.
But former No 1 Jankovic proved to be a tough customer, who levelled at a set each following an undisclosed change of tactics and left Bencic momentarily sobbing on her chair.
Jankovic will face German Andrea Petkovic in the second round. The Serb, who was runner-up 11 years ago in her Dubai debut, has also played four semifinals.
The veteran used years of experience to start turning the tables in the second set, with Bencic failing to profit from a lead of a set and a break.
With the Swiss serving to stay in the set while trailing 5-6, Jankovic failed on five set points before finally squaring the contest at a set each as Bencic drove a backhand wide.
The final set produced more frustration for Bencic, forced to save multiple break points in two service games and finally going down to defeat as she was broken in the last game after more than two and a half hours.
"I fought on every point, she was playing well," said Jankovic, who turns 31 at the end of the month.
"I changed strategy after the first set, it was that and my fighting spirit which got me through at the end. I'm so glad I was able to win."
