England were 76 for one at lunch after winning the toss and making first use of a pitch with good bounce but minimal sideways movement.
Cook made 27 in an opening stand of 55 with Alex Hales before edging Kagiso Rabada low to third slip where Morris dived far to his left and held the ball centimetres above the ground. Hales was 38 not out at lunch.
Rabada and off-spinner Dane Piedt were both playing in their fourth Test match, leaving Morne Morkel, in his 69th Test, as the only experienced member of the South African bowling attack.
Both Rabada and Morris built up good pace, with Rabada clocked at 150kmh shortly before dismissing Cook.
In a third change for South Africa, Quinton de Kock replaced out-of-form batsman JP Duminy. De Kock kept wicket, with AB de Villiers playing as a specialist batsman.
On a hot, sunny day, Cook and Hales started cautiously, scoring 11 runs off the first six overs before Rabada came into the attack and was hit for two fours off his first two balls by Hales.
Both batsmen were looking comfortable until Cook, after hitting four boundaries in a 49-ball innings was drawn into a drive against Rabada and Morris made his first significant contribution in a Test match.
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