When rain stopped play some 10 minutes before the scheduled interval, Cook was 34 not out and Westley 24 not out.
The Essex duo had put on exactly 50 for the second wicket after South Africa-born opener Keaton Jennings was out for a duck.
It was just the sort of 'correct' top-order Test-match display England's critics, including several former captains, had called for after the hosts were twice bowled out cheaply in a huge 340-run defeat at Trent Bridge that saw South Africa level the four-match series at 1-1 last week.
England had a setback soon after captain Joe Root won the toss and elected to bat despite the overcast conditions and green-tinged pitch that promised to assist a Proteas' pace attack bolstered by the return of Kagiso Rabada.
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The under-pressure Jennings was repeatedly beaten outside off stump by Vernon Philander and it was no surprise when he eventually edged the seamer low to Dean Elgar at third slip for a nine-ball nought.
But he immediately looked far more certain of where his off stump was then Jennings.
Westley confidently left his first two deliveries from Philander, whose opening spell of one wicket for two runs in four overs, including three maidens, was testament to accuracy.
The 28-year-old right-hander, given his chance after a finger injury saw the struggling Gary Ballance ruled out, got off the mark with a minimal risk legside four off Morkel.
Westley, who earlier this season scored a hundred against South Africa for the second-string England Lions, then cover- drove and on-drove Keshav Maharaj for two well struck boundaries in the left-arm spinner's lone over before lunch.