Djokovic, whose last defeat came against Rafael Nadal in the U.S. Open final, is aiming to be the first player to win four Australian Open titles in a row in the Open Era.
He is looking the part, despite being broken in his first attempt to clinch the match Friday.
"I have to be satisfied ... as the tournament progresses I'm playing better and better," Djokovic said. "It's important not to underestimate any opponent. Having the right approach to the court, having the right intensity, starting from the first point to try to show to your opponent that you're there to win."
The Spaniard was a semifinalist last year at Melbourne Park before losing to Djokovic.
In night matches, No. 15 Fabio Fognini beat American Sam Querrey 7-5, 6-4, 6-4 and will play Djokovic on Sunday. No. 17 Tommy Robredo defeated No. 9 Richard Gasquet 2-6, 7-5, 6-4, 7-6 (6) and to set up a fourth-round match against a well rested Stanislas Wawrinka.
Wawrinka received a walkover Friday when Canadian Vasek Pospisil withdrew with a back injury, giving him three full days off before his match against Robredo.
Ferrer, 31, had a career-best year in 2013, reaching his first Grand Slam final at the French Open and finishing the year at No. 3 in the rankings.
Dzumhur, the first man from Bosnia and Herzegovina to play singles in a Grand Slam, was cheered on by a vocal group of supporters from the Balkans region who now live in Melbourne.
Berdych made it to the quarterfinals last year, losing to Djokovic.
The No. 7 Berdych is again in Djokovic's half of the draw, but the two could not meet until the semifinals.
Information from The Associated Press was used in this report.