The 10-time champion has been struggling with illness and was well short of his best in his first two matches, particularly a second-round clash with Alexei Popyrin on Wednesday that he was lucky to escape.
But Djokovic, playing his 100th match at Melbourne Park, looked much more like his usual self in a 6-3 6-3 7-6 (2) victory over Argentina's Etcheverry, who beat Andy Murray in the first round.
The Serbian was hitting his groundstrokes with their customary ferocity again and, although there were perhaps still a few more mistakes than he would have liked, he was always fully in control of his 92nd win on these grounds.
Djokovic said: 'It was a great match, the best performance I had during this tournament. I'm pleased with the way I played throughout the entire match, particularly the first two sets.'
The form players of the first week have been fourth seed Jannik Sinner and big home hope Alex De Minaur.
Sinner finished last season beating Djokovic twice in one day as he led Italy to the Davis Cup title and has eased through to the fourth round here.
He dropped just four games in a 6-0 6-1 6-3 destruction of Sebastian Baez to set up a meeting with Russian Karen Khachanov, who was a semi-finalist here last year.
Sinner said: 'Today's match was really good. I played good tennis. I love the conditions also today, because was no wind and no sun.
'I felt great... I made the right decisions and I used every chance I had today, which maybe in the first match I was struggling a little bit. But it's good to have this match feeling back again. I'm very excited to step on the court every time.
'I'm doing my job. I try to stay focused. I try to find a solution against every player I play. The next round is going to be different, it's going to be tough.'
Khachanov, who has reached at least the quarter-finals at three of his last four slams, had a tougher time in beating Tomas Machac 6-4 7-6 (4) 4-6 7-6 (5).
De Minaur's rise into the top 10 just before the tournament fuelled hopes of a first home winner of the men's singles since Mark Edmondson in 1976, and he has dealt with the pressure immaculately so far.
The 24-year-old had no trouble getting past Italian qualifier Flavio Cobolli, dropping only seven games in a 6-3 6-3 6-1 victory.
Last year's runner-up Stefanos Tsitsipas also had a comfortable afternoon, beating young Frenchman Luca Van Assche 6-3 6-0 6-4 to set up a clash with American 12th seed Taylor Fritz, who came from a set down to see off Fabian Marozsan 3-6 6-4 6-2 6-2.
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