"I felt the pain but I didn't think about it", said after beating the Canadian.
Kerber continued her stunning start to 2018 by swatting aside Maria Sharapova 6-1 6-3 but Halep was detained for almost four hours by Lauren Davis in a remarkable 4-6 6-4 15-13 victory.
The tall Spaniard had a medical timeout at 2-5 in the first set and after having a blistered foot re-taped looked briefly regalvanised.
She needed only 59 minutes to end the campaign of 2017 Australian Open junior champion Kostyuk, who committed 31 unforced errors, compared with only 11 from the racquet of Svitolina.
Kerber will now play in the fourth round with either Agnieszka Radwanska or Hsieh Su-wei.
Karolina Pliskova and Madison Keys, seeded sixth and 17th respectively, preserved their flawless form with straight-set wins in Round 3, too; neither player has dropped a set at the tournament yet. Clearly, the 2014 champion was too ambitious.
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"If you haven't played for a long time, anything that isn't quite the centrepiece of your game, can go through a bit of a challenge", former world No. 1 Agassi, who flew in to be in Djokovic's corner, told AAP on Friday. "To be that far already, it's more than what we could have expected".
Still, he added: "It's a huge deal for me, and something that if I don't ever get another accomplishment, I can at least hang my hat on this one". He won in Milano previous year.
The Frenchman remarkably recovered to make a fight of it before a relieved Djokovic , seeded 14, came through 4-6, 6-3, 6-1, 6-3.
Afterwards, he called the conditions "risky" for the players, especially with the new rule restricting the time between points to 25 seconds.
Halep saved three match points in the 22nd game of the third set at 0-40 and Davis saved five break points in the 23rd game in the 3 hour, 45-minute match.
"It was obvious we both suffered on the court today. It was brutal for the first hour and a half", said the Serb. Novak Djokovic, returned from his six-month injury break with wins over Donald Young and Gael Monfils in Melbourne, faces a tricky encounter in the No.21 seed Albert Ramos-Vinolas. It was just one of these days where you had to stay tough mentally. It has always been, not just for tennis, but in general there's always something that you can improve and get better at as a person and as a player.