Having lost the Test series 2-0 and fallen 1-0 behind after two matches of the one-day leg, England turned the tide at the Kia Oval courtesy of Morgan's fluent 73 and a stoic 71 from Trott.
The bowlers earlier played their part by dismissing the tourists for 211, Ravi Bopara miserly with one wicket for 31 runs in 10 overs and James Anderson polishing off the tail to close with four for 44.
England briefly looked in bother on 64 for three but Trott played the anchor role in a dogged 125-ball innings, while Morgan added the flair with seven fours and two sixes.
Ian Bell wasted no time getting England's chase going but, after crashing three boundaries in Lonwabo Tsotsobe's first over, he was pinned leg before wicket by Dale Steyn.
His opening partner Alastair Cook took 18 balls to get off the mark but did so by turning Tsotsobe off his hips for four.
Eleven more came off Steyn's fourth over as Trott arrived at the crease and the hosts reached 45 for one after 10 overs.
Tight lines from Morne Morkel and Wayne Parnell saw just 12 runs added in the next five overs.
Cook was particularly sedate and went for a laboured 20 with a tame shot to Robin Peterson.
Bopara then fell for a disputed duck, given caught behind off Morkel.
He immediately reviewed the verdict but the technology was inconclusive and he was forced to depart, albeit reluctantly.
At 69 for three from 20 overs, England had plenty of work to do.
Morgan's arrival added much-needed impetus - not to mention boundaries - as he raced past 30 at better than a run-a-ball.
A wide from Dean Elgar took England's target below 100 and Morgan thrashed the spinner for the first six of the day in the 32nd over.
All the while, Trott was compiling slowly but surely and reached 50 from 86 deliveries.
Morgan's own half-century came off a breezier 54 balls and he overtook his partner with a flowing four over extra-cover.
A second six followed to the same area off Tsotsobe and his seventh four, off Morkel, brought up the century stand.
Morgan's game-changing knock ended with a looping caught-and-bowled to Peterson and 40 still needed.
Craig Kieswetter cleared the ropes once before being run out for 14, leaving Trott to shepherd England within five runs of victory.
The impressive Parnell accounted for the number three, via the outside edge, but Samit Patel was on hand to hit the winning boundary.
South Africa, having won the toss and batted, got off to a promising start but after reaching 119 for two at the halfway stage they lost wickets at regular intervals to be bowled out with 3.2 overs remaining.
Hashim Amla, having made a 311 not out on this ground in the Test series, was quickly into his stride and plundered 11 off James Anderson's third over.
His partner, Graeme Smith, had 18 of the first 50 runs before he was bowled by Anderson to an ugly stroke.
Cook turned to Bopara and his nagging medium pace soon after and he successfully strangled the scoring.
It was Jade Dernbach who made the key breakthrough, though, swinging a full delivery past Amla's bat and removing leg stump for 43.
The bowling powerplay boosted England, costing just 21 runs and yielding that prize scalp.
Elgar and AB de Villiers were in danger of getting stuck until they took 13 off a single James Tredwell over, although the off-spinner exacted his revenge at his next visit.
De Villiers was the man out, chipping to Bell at long-off for 28.
Bopara got a deserved wicket when he bowled Faf du Plessis for one and Elgar went for a less-than-fluent 42 when he missed a trademark Dernbach slower ball.
Number seven Parnell had a frenetic eight-ball stay, hitting one authentic four through midwicket and two edged boundaries before Dernbach had him caught behind.
That left JP Duminy to repair the innings and he made a diligent 33 before holing out off the bowling of Tredwell looking to up the rate.
Anderson made light work of the lower order, removing Steyn, Morkel and Tsotsobe in the space of eight deliveries.
- James Anderson