Central Coast forged ahead in the 36th minute when debutant Milan Duric's thunderous strike from 25 yards crashed in off the underside of the crossbar.
But its lead lasted just five minutes, with Duke heading in Pirmin Schwegler's corner at the near post to restore parity.
Duke sealed all three points in the 82nd minute, slotting home from the penalty spot after Dylan Fox was been penalised for handball after a VAR (video assistant referee) review.
The ex-Shimizu S-Pulise striker had the final say in stoppage-time, clearing Ziggy Gordon's effort off the line after the defender's initial header had been pushed back to him by goalkeeper Daniel Lopar.
Elsewhere, Melbourne Victory and Melbourne City played out an uninspired goalless draw.
Victory came closest to securing the win at Marvel Stadium, but Elvis Kamsoba's goal 10 minutes into the second half was ruled out for offside by VAR.