The indomitable Serb second seed avoided a potential first-round exit against Verdasco, who knocked out Rafael Nadal in last year's first round.
Djokovic prevailed 6-1, 7-6 (7/4), 6-2 in 2hr 20min on Rod Laver Arena to progress to a second-round encounter with Uzbekistan's Denis Istomin.
"I'm very pleased with the first round, considering I had one of the toughest first-round draws, definitely considering his form, how well he played against me in Doha," Djokovic said.
"I think that I could have drawn an easier player.
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It was a tricky first-up match for the 12-time Grand Slam champion, who had lost to the Spaniard in four of their previous 13 meetings.
But Djokovic's greater consistency and composure in the pressure moments of an intense match won him the day.
"I think from one perspective it was good that I got to have the very tough first-round match," he said.
"It made me prepare better and approach this match and the tournament with the right intensity right from the blocks, right from the first point."
Djokovic is seeded to face Brisbane International winner Grigor Dimitrov in the fourth round, powerful Austrian eighth seed Dominic Thiem in the quarters and Canada's big-serving third seed Milos Raonic in the semi-finals in the bottom half of the men's draw.
He can take outright ownership of the record for Australian titles if he wins for a seventh time. The Serb is currently tied with Australia's Roy Emerson on six.
Both players traded service breaks in four straight games before the second set went to a tiebreaker. Djokovic's greater consistency and Verdasco's erratic groundstrokes gave the Serb the tiebreaker and a 2-0 sets lead.
"He was on top of it. I just didn't want to give him too much of a chance to step in and kind of dictate the play. In second set, it was very close," Djokovic said.
Djokovic cranked it up in the third set, breaking Verdasco's opening service which he carried through to win on his first match point.
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