Although Hartono, 29, improved as the set progressed – she hit 13 winners compared to her opponent’s four – the Dutchwoman was made to pay for her 13 unforced errors.
The second set went with serve until the third game, when Jones – top seed in the qualifying draw – fought back from 40-15 down to secure her third break of the day.
World number 187 Hartono immediately hit back with a break of her own as Jones became a little passive, but the Briton tightened her footwork and broke once again to wrestle back control of a topsy-turvy set.
She finished the match with 10 consecutive points on serve before she sprinted over to her team in celebration.
An increased prize pot meant the Bradford-born world number 86 has won $110,000 (£81,238) for reaching the first round.
