England had crushed the tourists by nine wickets in the first match of the series in Southampton.
However, following Jason Roy's controversial dismissal for obstructing the field, the momentum changed as England quickly went to 171-6, eventually losing by three runs.
But skipper Morgan repeated after Wednesday's opening T20 thrashing of South Africa that if Roy stays at the crease for any length of time then the inevitable conclusion is an England victory.
Late hitting from Farhaan Berhardien, who made 32 from 21 balls saw the Proteas make 174-8 from their 20 overs despite Curran removing Morris for 12 and Phehlukwayo for a first ball duck.
Roy's first half-century in 11 internationals was a decent response to being dropped for last week's Champions trophy semi-final against Pakistan.
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South Africa made just the one change, with leg-spinner Imran Tahir coming in for left-arm wrist-spinner Tabraiz Shamsi.
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The decision was made by the third umpire, Tim Robinson, who decided that Roy deserved to be dismissed having looked at the fielder, Andile Phehlukwayo, and crossed the pitch to intercept...
South Africa's Chris Morris was named man of the match for his decisive four-over spell of two for 18.
England got off to the flawless start thanks to Curran, who had an instant impact after coming on in the fourth over to bowl Reeza Hendricks with his second ball in worldwide cricket to leave South Africa on 25 for one.
Asked if those were the darkest conditions he had played in, Morgan replied: "By a country mile".
Earlier, Cape Town-born Tom Curran, a Surrey team-mate of Roy, marked his England debut with an impressive three for 33 in his maximum allowed four overs.
But Mosehle (15), after breaking his second bat as well, was caught behind gloving a slower ball from paceman Liam Plunkett (two for 36). AB de Villiers (captain), 5.
On-field umpire Michael Gough then signalled a dead ball while the soft signal was not out. The tenth over, bowled by Plunkett, was dispatched for 21 runs which included a beautifully timed six over cover, perhaps the shot of the innings.
Elsewhere in the England side, former Somerset star Jos Buttler will keep wicket as he returns to play at the County Ground for only the second time since leaving for Lancashire in 2013. Smuts kept playing aggressively - he hit two sixes off spinner Liam Dawson, one of which landed on the pavilion roof - and moved to 45 from 35 balls before somehow hitting a full-toss from Dawson straight up in the air.