The Eels had to withstand a fightback from the Cowboys, who scored two late tries to roar back from 14-0 down.
Last year's losing grand-finalists, Parramatta have had a miserable start to the season, winless through five rounds to sit at the bottom of the ladder.
But they held their nerve to hang on and secure the win at Netstrata Jubilee Stadium on Saturday.
Hunt overcame a shaky start to score a try in each half, with both tries coming off the boot of halfback Rachael Pearson.
The Cowboys looked down and out until debutante Libby Surha charged over the line in the 59th minute, followed by Tiana Raftstrand-Smith two minutes later to set up a thrilling finale.
Surha was forced off the field with just over a minute remaining after appearing to dislocate her shoulder.
Hunt's first try came at the 32nd minute mark when Pearson produced a precision cross-field kick, with the winger diving across in the corner.
The halfback then slotted the conversion from the sideline to give Parramatta a 10-0 halftime lead.
Straight after halftime the pair combined again with Pearson putting up a kick over the head of Cowboys winger Krystall Blackwell for Hunt to collect her second.
North Queensland, who sit one spot above the Eels in ninth spot, were denied an early try by China Polata after the bunker ruled obstruction.
Hunt missed out on another opportunity when a long pass from Mahalia Murphy to the waiting winger went forward.
Second-rower Murphy opened the scoring in the 20th minute with the Test rugby union star dragging three defenders over the line.
Pearson capped the win with a penalty kick in the final seconds, before she was swamped by her celebrating teammates.
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