Riding the momentum from her fourth title of the season last week at her home tournament, the Warsaw Open, Swiatek looked poised for easy passage breezing through five straight games breaking Muchova twice to claim the first set.
But in each of their two previous clashes the contest went three sets and the third encounter would be no different as the Czech charged ahead 4-1 in the second and then after a little wobble, coolly closed out the set to level the match.
Just as play was about to begin in the third, rain swept across Montreal sending players to the locker room.
They returned and Swiatek broke to open the deciding set but rain once again forced a delay.
Later, Swiatek would hold for 2-0 and that would be all the cushion she would need to win the rematch of the French Open final, where Swiatek also prevailed for her fourth Grand Slam title.
"You have to find energy even though we've been here since 9 a.m.," Swiatek said after the match, which took more than nine hours to complete.
"For sure, it was a pretty extraordinary day, and I don't think I've had such a situation in my career so there is the opportunity to learn something new and see what I'm capable - even though we played this match like three times."
With the win, Swiatek guaranteed that Aryna Sabalenka, whose round of 16 match was postponed until Friday due to the weather, will not overtake her at the top of the rankings come Monday.
Swiatek will face American qualifier Danielle Collins, who dispatched Canadian wild card Leylah Fernandez 6-2 6-3, for a spot in the semis.
Earlier, fourth seeded American Jessica Pegula romped into the last eight with a 6-4 6-0 thrashing of Italian Jasmine Paolini, who in five career meetings has failed to take a set off the American.
The two women traded early breaks before Pegula seized control with a break to claim the opening set then shifted into top gear steamrolling through the second to ease into her 10th quarterfinal of the season second only to Swiatek.
Pegula will battle her doubles partner and compatriot Coco Gauff in the quarters after the teenager crushed Wimbledon champion Marketa Vondrousova 6-3 6-0 under the lights in Montreal.
