The Cape Verdians took a shock but deserved lead in the 36th minute through Platini only for Morocco to score in the last 12 minutes.
However, with two points from their opening two Group A games, Cape Verde - with a population of just over 500,000 the smallest country to ever compete at the tournament - still have a chance of progressing to the knockout stage.
They face Angola in their final group match in Port Elizabeth on Sunday while Morocco meet hosts South Africa in Durban.
"It's a night to congratulate Cape Verde football, a big night for our country," said coach Lucio Antunes.
"We showed we have great players just like Ghana, Nigeria, the Ivory Coast. My boys can play at Chelsea, Manchester United, Milan, Real Madrid," he added as emotions threatened to run away with him.
Cape Verde hussled and fought throughout the game. Their midfield pressure forced Karim El Ahmadi to lose the ball, allowing Ryan Mendes to run away and set up Platini for the opening goal, which he chipped over on-rushing Morocco keeper Nadir Lamyaghri.
It was one of several missed chances for the Cape Verde, who also had a thunderous free kick from Mendes saved by Lamyaghri.
Morocco did not come alive until late in the game but did enough for a draw when the unmarked El Arabi drove the ball home after a pull back from Abdelaziz Barrada in the 78th minute.
It sparked Morocco, a country with a long tradition of success in the African game, out of a collective slumber and set up a grandstand finish with Al Arabi missing a chance to grab the winner in the 83rd minute when his header went wide.
South Africa lead the group after beating Angola 2-0 on Wednesday. They have four points. Cape Verde and Morocco are on two and Angola one.
- Sports & Recreation