Burkina Faso and Ethiopia also came through suspense-filled matches to book their places and were accompanied by heavyweights Nigeria and Senegal.
The five countries now compete over the next two months in play-off games to determine Africa's representatives for next year's finals in Brazil.
Algeria, Egypt and Ivory Coast secured their berths back in June, Ghana did so on Friday and the last remaining place will be decided on Sunday when Cameroon host Libya with the home team needing a draw and the visitors a win.
The draw for the play-offs is in Cairo on September 16.
First-half goals from Platini, who takes his name from the former France captain, and Heldon produced a giant-killing Group B result in Tunis for one of Africa's smallest countries who began the year with a shock run to the Nations Cup quarter-finals.
Tunisia had needed only a draw to go through.
Burkina Faso, surprise runners-up at the Nations Cup, completed a stunning comeback after losing their opening two games as they edged Congo to the top spot in Group E.
Congo had led by a point but were held 2-2 in Niger despite drafting tall centre back Christopher Samba into their defence.
It opened the door for the Burkinabe to overtake them and Prejuce Nakoulma scored nine minutes into the second half to earn a 1-0 home win over Gabon that took them to 12 points, one more than Congo.
Ethiopia were on their way out of the competition at halftime against the Central African Republic but rallied to win 2-1 in neutral Brazzaville and finish top of Group A.
It was a relief for Ethiopia who were previously stripped of three points when soccer's ruling body FIFA found them guilty of using a suspended player.
The deduction left them needing to win on Saturday to stop South Africa snatching a dramatic lifeline.
South Africa were 2-0 up at home to Botswana at halftime and Ethiopia down by a goal, which would have seen the 2010 World Cup hosts through on goal difference.
But second-half goals for Ethiopia by Saladin Seid and Minyahile Teshome left South Africa crestfallen despite their 4-1 triumph in Durban.
Nigeria and Senegal needed home draws to win their respective groups but won to emphasise their superiority.
Emmanuel Emenike, returning after an eight-month injury absence, opened the scoring for Nigeria and Victor Moses added a second-half penalty as the African champions beat Malawi 2-0 in rainy Calabar to finish first in Group F.
Sadio Mane netted six minutes from time as Senegal defeated Uganda 1-0 in a scrappy Group J match played in neutral Marrakech.
Uganda had Godfrey Walusimbi sent off in the first half for a wild tackle.
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