It was a comfortable victory for Britain's Murray and his Brazilian partner - winning their match 7-5 6-2 6-1 in an hour and a half.
"Just ran out of time in the end", Williams said.
Venus couldn't make it a third consecutive comeback triumph as she lost to 20th-seeded Kiki Bertens of the Netherlands 6-2, 6-7 (5), 8-6 in the third round on Friday.
Compatriot Joe Salisbury is also through to round three with his partner Frederik Nielsen after their opponents Rohan Bopanna and Edouard Roger-Vasselin retired two sets and 3-2 down in the third. "I had no idea what my draw was and all of a sudden I came in here the other day, it was like, So if you win, then you play this person [Serena]".
Asked if nerves got the better of him, last year's beaten finalist Cilic said: "I was still focusing on my game; it was nothing in my mind that I have to go far or that I'm obliged to do that". Another ace, her 12th, gave her match point and a 13th the match.
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John Isner, the U.S. ninth seed, saved two match points, fired 64 aces and 88 winners to defeat Belgian qualifier Ruben Bemelmans 6-1, 6-4, 6-7 (6/8), 6-7 (3/7), 7-5.
Rodina won nine games in a row to move a set and two breaks ahead before Keys recovered from 4-0 down in the second set to set up a decider.
Isner will face world No.31 Stefanos Tsitsipas, the first Greek man to first Greek man to reach the fourth round of a grand slam in the 50-year professional era, for a quarterfinal berth.
Nick Kyrgios, the 15th seed, also advanced in straight sets, beating Dutchman Robin Haase 6-3 6-4 7-5 to book a date with Kei Nishikori, who ousted Kyrgios's fellow Australian Bernard Tomic 2-6 6-3 7-6 (9-7) 7-5.
The Frenchman reached the semifinals at the 2008 French Open and the 2016 US Open.