The 32-year-old will bid for the quarter-finals in Toronto for the first time since 2015 when he plays Stan Wawrinka, who staged another fightback in three sets, beating Marton Fucsovics 1-6, 7-6 (7/2), 7-6 (12/10).
Kyrgios and Wawrinka, while enjoying an amicable relationship now, were embroiled in a dicey moment in Canada three years ago when the Aussie alleged that countryman Thanasi Kokkinakis had slept with Wawrinka's girlfriend and WTA Tour player Donna Vekic.
The International Tennis Federation's controversial revamp would see the Davis Cup morph into a World Cup of Tennis final featuring 18 countries - and address complaints that the heavy time commitment drives away the top players.
The ninth-seeded Serb, competing in his first tournament since his Wimbledon triumph last month, won a dominant 83 per cent of his first service points.
This was after the Spaniard was made to wait before his second-round match due to several rain delays.
The decider ended in a 22-game tiebreak, with the former world No 3 taking the set 7-6 (12-10).
Teenaged prodigy Shapovalov eliminated Italian 14th seed Fabio Fognini victor last weekend in Los Cabos, 6-3, 7-5.
When Kyrgios was 4-1 up in the first set, with Wawrinka serving, the Aussie absolutely nailed a forehand victor that left his three-time Grand Slam-winning opponent grasping at thin air and applauding the shot. World number four Kerber, however, slumped to a 6-4, 6-1 loss to Alize Cornet of France.
The inclement weather also forced Shapovalov to wait, but the Canadian had no issues dispatching Chardy 6-1 6-4.
The two-time Australian Open champion Azarenka eventually rolled through the final three games to get the victory in her first Montreal appearance since 2014.
"No matter what tournament or who you're playing against, you have to figure things out as you go".
"I thought I served well in the moments when I really need it and I was looking for the first serve", said Djokovic.
Sharapova, a finalist in 2009, performed solidly, defeating Karatantcheva for the fifth time in their five career meetings. "I never expected to beat such a top-ranked player and such a high-respected player like Novak".
He was replaced by the 27-year-old Bosnian who was outclassed early, but troubled the 4-time Canadian champ in the second, breaking him twice late, before falling.
The Russian will face sixth-seeded Caroline Garcia of France in the third round.
Djokovic, coming off his fourth Wimbledon title, beat Canadian wildcard Peter Polansky 6-3, 6-4.