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Zverev edged Dutchman Tallon Griekspoor 3-6 6-4 6-2 4-6 7-6 (10-3).
Defending champion Djokovic outclassed Spaniard Roberto Carballes Baena 6-4 6-1 6-2.
Ruud swept aside Brazil's Felipe Alves 6-3, 6-4, 6-3 while Sabalenka mowed down Russia's Erika Andreeva 6-1, 6-2.
The title was the 21st of Swiatek's career as the world No.1 extended her unbeaten run to 12 matches on clay to sound out a warning to her rivals ahead of the French Open.
Victory improved Swiatek's record over her Belarusian rival to 7-3 as the three-time French Open champion bagged the only major claycourt title that was missing on her glittering resume.
After a gutsy second-round win on Friday over good friend Paula Badosa the world No. 2 appeared to have little left in the tank physically, mentally or emotionally for a late night Stadium court battle with Ukrainian Kalinina.
The statement was the first mention since the tragedy that the couple had split.
Forward Koltsov played for the Belarus national team in the 2002 and 2010 Winter Olympics and spent parts of three seasons with the National Hockey League's Pittsburgh Penguins from 2002 and 2006.
Navarro, seeded 23rd, broke the hard-hitting Belarusian for a fourth time on match point to book a quarterfinal clash with ninth seed Maria Sakkari of Greece.
The Belarusian second seed is the first woman to retain the Melbourne Park title since compatriot Victoria Azarenka in 2013.