Patchell had landed two penalties and a conversion of a penalty try before, having missed a conversion of wing Tom James' late try, he booted a touchline three-pointer to seal a 21-20 victory.
The Dragons had been 17-6 up during the second half after wing Ashton Hewitt scored a first-half try and fly-half Dorian Jones booted three penalties and a drop goal, with Jones' replacement Jason Tovey kicking a fourth penalty.
The win sends the Blues above the Dragons into ninth place in the Guinness PRO12 table.
Edinburgh sent Glasgow packing from the capital for the second season running as the Gunners took a step towards reclaiming the 1872 Cup with a 23-11 first-leg win.
In front of a record inter-city crowd of 23,642, the hosts at Murrayfield fell behind to Mark Bennett's early try but hit back to win with scores from Matt Scott and John Hardie.
With Guinness Pro12 points also on offer, Alan Solomons' team now leap-frog Glasgow into the top six ahead of next Saturday's Scotstoun return.
Edinburgh's victory in the annual festive double-header 12 months ago ended Glasgow's five-year reign of dominance and injected a dispirited squad with a large dose of self-belief.
An 83rd-minute try from Dries van Schalkwyk handed Zebre a last-gasp 28-25 win over Italian rivals Benetton Treviso in the Guinness Pro 12.
Zebre led 16-3 at half-time and looked comfortable but Benetton mounted a comeback and a converted try from Matteo Zanusso five minutes from time appeared to have won it.
But there was one final twist and Van Schalkwyk's score, converted by Kelly Haimona, gave the hosts the win and kept them above Benetton at the bottom of the table.