Goals from Callum McGregor, Giorgos Giakoumakis, Daizen Maeda and James Forrest sealed a comfortable win for the holders and set up a last eight clash away to Motherwell, who also scored four in knocking out Inverness Caledonian Thistle.
The draw also handed Rangers a home tie against Championship side Dundee, while Kilmarnock, who stunned Hearts with a 1-0 win at Tynecastle, will host Dundee United, who rebounded from their 9-0 defeat to Celtic and the sacking of manager Jack Ross with a 2-1 win at Livingston.
Aberdeen will welcome Partick Thistle to Pittodrie in the remaining quarter-final.
Celtic made nine changes for the trip to Dingwall and Postecoglou was pleased with the response as he now turns his attention to Saturday's first Old Firm fixture of the season against Rangers.
"The focus we showed really pleased me most," he told Premier Sports. "A lot of the guys were playing their first games and we didn't let our performance level drop at all.
"We haven't looked beyond today, but now that today is done we'll get a recovery. We won't get back until late to Glasgow but we'll still train tomorrow, do our normal processes and look forward to Saturday."
Asked whether any further incomings or outgoings could be expected before tonight's 11pm transfer deadline, Postecoglou added: "Yeah, there's some things happening but I'll be on a bus for the next few hours so I'll find out when I get off."