The visitors, though, had a chance to tie the scores and send the match to a 'super over' when paceman Reece Topley fluffed a straightforward opportunity to run out Kyle Abbott off the last delivery.
Having earlier compiled a modest total of 134 for eight, England strangled the run-rate on a wicket that both sides found difficult to score on and looked set to win after leaving South Africa needing 15 runs from Topley's last over.
Number eight batsman Morris, however, clubbed a six and a four to lead the home team over the line.
The all-rounder also hit a swashbuckling half-century in the fourth game of the one-day series last Friday to help his side scramble a one-wicket triumph.
England were put in to bat in the opening match of a series of two Twenty20s and blazed their way to 50 in six overs for the loss of opener Jason Roy (15).
The in-form Alex Hales (27) was then caught by JP Duminy at deep square-leg, despite a sickening collision with a charging Kagiso Rabada that thankfully left both fielders unhurt, and his dismissal sparked a slump to 81 for six.
Leg-spinner Imran Tahir was the chief destroyer for South Africa as he equalled career-best figures of four for 21.
Jos Buttler top-scored with an unbeaten 32 from 30 balls.
Captain Faf du Plessis (25) led South Africa's reply from the front but excellent bowling spells from Chris Jordan (3-23) and Moeen Ali (2-22) appeared to swing the contest England's way until Morris produced his late cameo.
The second game in the series, which is serving as a warm-up for the World Twenty20 in India next month, is in Johannesburg on Sunday.
(Reporting by Nick Said; Editing by Tony Jimenez; nick.said@thomsonreuters.com; +27832722948; Reuters Messaging: Reuters Messaging: nick.said.thomsonreuters.com@reuters.net)
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