The veteran striker, who joined Oldham half an hour before the transfer window closed, gave his new side an early lead but Anthony Kay hit an equaliser.
Kuqi needed just six minutes to make his impact after Bradley Diallo produced a fine run-and-cross, setting up the Finn to convert a six-yard header.
But Huddersfield levelled after 18 minutes when Gary Roberts took a left-wing corner and Kay blasted an unstoppable volley in off the post from 12 yards.
Filipe Morais tested Huddersfield goalkeeper Ian Bennett from 35 yards and, just after the break, Bennett made a superb reflex save from Reuben Reid.
Oldham had Diallo sent off in the 76th minute, the ex-Marseille man shown a straight red card for his scything challenge on Oscar Gobern, and Huddersfield substitute Alan Lee nearly won it late on but his shot came back off the bar.