The number three seed fought back on two occasions, winning four straight games from 3-5 down in the first set and then recovering from a 2-6 deficit in the second set tie-break on the way to a 7-5 7-6 (9/7) victory.
The 20-time grand slam champion will stay second in the rankings if he wins the title and Djokovic fails to reach the final.
After his victory, he said: "It was extremely hard, that is why I was very happy at the end".
Zverev has been suffering with a cold but ratcheted up the pressure to grab the all-important break of serve in the ninth game before closing out the match against de Minaur.
"I didn't play many poor matches, to be quite honest".
Others, while condemning Verdasco's behaviour, pointed out that he is by no means the first tennis player to treat a ball kid harshly.
The top seed and 20-time Grand Slam victor emerged from another nervy battle to eventually defeat the 28th-ranked Spaniard 6-3, 2-6, 6-4. "At this point in my life, I want to do what I really like to", said Federer. "I do not want to paint myself as a ideal person".
China's leading ladies will be bidding to join Muguruza in the quarter-finals when they play on Thursday. "I'm just really happy".
Thiem, who also received a first-round bye, struggled to find his rhythm in a tightly contested encounter which lasted two hours and 45 minutes.
"Li Na has given us all a lot of hope", Zhang said.
Alexander Zverev of Germany watches his shot as he plays against Nikoloz Basilashvili of Georgia during their men's singles match in the Shanghai Masters tennis tournament at Qizhong Forest Sports City Tennis Center in Shanghai, China, Wednesday, Oct. 10.
"I think all the top guys will say the same thing". We have one and a half months off in our season, and that's end of November and December.