King, who reached the semi-finals at Alexandra Palace in 2009, will now face either Colin Osborne or Michael van Gerwen in the second round after Christmas.
King started the match confidently with a 180 in the process of taking the first leg and doubled his lead by landing double 14.
King, who defeated Phil Taylor in the Cash Converters Players Championship a week ago before losing to eventual winner Kevin Painter, hit double eight in the third to claim the opening set comfortably.
King, seeded 12 for this year's World Championship, hit double four to make a positive start to set two and then won the next leg with the aid of his third maximum.
Kime was struggling to find any rhythm on his big stage debut, and King won his sixth successive leg by hitting double top to establish a two-set lead.
Kime - who was born in Leeds before moving to Australia aged 12 - missed two darts at doubles to take his first leg of the match at the start of the third set, and King hit double eight to edge ahead.
King won the second before Kime hit a 180 in the third, but he missed three chances at doubles and his opponent hit double ten to end the game.
Oceanic Masters winner Kime admitted: "I'd practised well and felt good going onto stage but it took me too long to settle. It's a great experience and I'd love to return here again the future."
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