Germany's Kerber was downed 7-5, 6-1 by Australia's Ashleigh Barty, last year's runner-up in Wuhan.
Wozniacki, the No. 2 seed, however, couldn't really get going as she stumbled to a stunning 6-7 (10) 5-7 defeat to her opponent from Puerto Rico, who came all the way from the qualifiers.
Halep, who injured herself in training earlier this week, offered some resistance in the second set after a bad start to the match, but could not prevent an early exit in her first tournament since the US Open.
Puig had come from a bagel set down to defeat the Australian Open champion 0-6 6-4 6-4 in the second round at the Miami Open in March and the 24-year-old repeated the dose with a forehand victor to set up a clash against home favourite Wang Qiang, who defeated Daria Gavrilova in straight sets, in an all-unseeded battle.
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The 22-year-old Barty said she is looking to sustain that momentum to end 2018 on "the best note possible, whether that is results-based or feelings-based".
The Australian started the match strongly, racing into a 5-2 first set lead.
The Aussie wrapped up the match in just over 75 minutes, setting up a quarter-final against Anastasia Pavlyuchenkova. I feel like I'm a top-20 player.
"It was tough to play in the beginning because my back was blocked", she said. I pushed myself to play better.
Barty will next face Anastasia Pavlyuchenkova for a spot in the semi-finals after the Russian was a surprise victor over fifth seed Petra Kvitova.
Tomic, who had his ranking bottom out at No.243 in May, saved a match point on Wednesday against South African Lloyd Harris in a 7-6 2-6 7-6 win.