Cook reminded the hosts of his ability with a solid knock of 126, bringing up his century with a six over mid-wicket, while Root scored 117 as England amassed 414 for six before declaring late in the day.
Stuart Broad and Moeen Ali claimed a wicket each as South Africa A reached 35 for two in their second innings, 243 behind the tourists, after the second day of the three-day match.
The contribution from Cook's opening partner Hales will have gone a long way to assuaging concerns about handing him a test debut on Saturday, when England and South Africa begin their four-match series in Durban.
Hales had been pencilled in to open with Cook after showing form in limited overs internationals but after scoring just 16 runs in two innings in England’s opening warm-up game in Potchefstroom last week, he needed a good knock to back up the confidence shown in him by coach Trevor Bayliss.
Cook and Hales compiled a confidence-boosting opening partnership of 112 before Hales was bowled by an in-swinger from Dane Paterson at the City Oval.
Nick Compton, who is expected to make his test return at number three for England, scored 29 before being trapped leg before wicket.
Cook went when he lobbed spinner Keshav Maharaj to mid-on but England did not let up on the home bowlers as Root raced to 50 off 60 balls and brought up his century off 116 balls. He was out when he top edged a pull to backward square and was caught off Maharaj’s bowling.
Root and Ben Stokes, who scored a brisk 66, put on 104 runs for the fifth wicket.
The final day's play will provide an opportunity for the bowlers to stake a last claim for test selection with Steven Finn and Mark Footitt looking to impress.
(Reporting by Mark Gleeson in Cape Town; Editing by Justin Palmer; mark.gleeson@thomsonreuters.com; +27828257807; Reuters Messaging: Reuters Messaging: mark.gleeson.thomsonreuters.com@reuters.net)
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