His opening partner Alastair Cook took 18 balls to get off the mark and 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 Morne Morkel but Morgan's arrival added much-needed impetus - not to mention boundaries - as he raced past 30 at better than a run-a-ball.
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.
Jade Dernbach offered support to Anderson with three for 44 and the in-form Hashim Amla top-scored with 43 for the Proteas.