It was their earliest meeting since a 17-year-old Venus beat 16-year-old Serena in the second round of the 1998 Australian Open.
Returning after more than a year away from the game to have a baby, Serena put up a good fight, saving match point while trailing 5-2 in the second set, but her rustiness told in the end. "So quite simple rule".
By reaching the quarterfinals, Kasatkina is projected to surpass her career-best ranking of 17th.
Cilic won 19 of 23 points on his first serve, but failed to convert four break points against 31st-seeded Kohlschreiber, who snapped a 12-match skid against top-10 opponents.
"So I have just got to keep saying at each tournament that my goal is just to be better than the last".
Rising Russian Daria Kasatina added to the misery for luckless USA open victor Sloane Stephens, handing the slumping American her tenth loss in 13 matches since winning the grand slam title in September, 6-4 6-3.
Her lateral movement was surprisingly good. In her postmatch media conference, she said she started listening to Smokey Robinson & The Miracles a couple days ago, discussed the meaning behind the name of her clothing line - "Life is about not having any confinements. I think it's such a treat for everyone to see this match so early in her comeback", Venus said. I definitely don't want to go backwards.
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"I'm getting pretty confident at this level".
She will face Angelique Kerber who crushed seventh seed Caroline Garcia in a 6-1 6-1 triumph. Serena went on the win the US Open title.
Halep advances to the quarter-finals where she will face unseeded Petra Martic of Croatia who beat Czech Marketa Vondrousova 6-3, 7-6 (7/4). She came up short on some shots that she would have made in her prime. "I really already was resetting, because the tournament is far from over". "Hopefully, she decides to play a lot and enjoy herself in the process and get back to who knows how high?" I have many things to learn from them.
"[Serena's] level is super high and it was very hard to close out the match, just getting one ball back", Venus said. Serena returned to the tournament in 2015 and Venus a year later.
At one point he won 17 straight points en route to taking a 6-2, 3-0 lead, winning 10 of the final 11 games in the match.
"It was not an easy match to figure out", Venus said of her victory Tuesday.
"It was a quick turnaround from last night when it feels like, 'Hey, this is a final to you're only in the fourth round, "' said Williams, as revealed by the New Zealand Press-based press.
It is rare for the Williams sisters to play this early in a tournament.