Platini and Nhuck — two of many Cape Verdean footballers who use nicknames — scored in the first half to trigger celebrations in the islands state of 500,000 inhabitants.
Victory lifted the ‘Blue Sharks’ to 12 points, one more than four-time World Cup qualifiers Tunisia, and completed a remarkable Group B comeback.
The surprise 2013 Africa Cup quarter-finalists were pointless after three rounds, then FIFA changed a loss in Malabo to a 3-0 win because of an ineligible Equatorial Guinea player.
Home victories against the Equatoguineans and Sierra Leone brought the Cape Verdeans back into contention — but they had to win in Tunisia.
The ‘Carthage Eagles’ were favoured to qualify for the play-off series as they had home advantage and the confidence boost of a 2-1 win in Praia.
But goals from Cyprus-based Platini and Portugal-based Nhuck rocked the Tunisians, whose second-half pressure went unrewarded.
Tunisia were not the only ‘big name’ to fall — 2010 World Cup hosts South Africa bit the dust despite a 4-1 win over Botswana in Durban.
Bottom Group A seeds Ethiopia came from behind to edge Central African Republic 2-1 in Congo and finish two points ahead of Bafana Bafana (The Boys).
Burkina Faso completed an amazing Group E comeback by defeating Gabon 1-0 in Ouagadougou and pipping Congo Brazzaville.
The ‘Stallions’ trailed the ‘Red Devils’ by six points with three rounds to go, but three consecutive victories squeezed them through.
Nigeria were another country to progress, scoring late in the first half and early in the second in Group F to defeat 10-man Malawi 2-0 in Calabar.
Cape Verde, Ethiopia, Burkina Faso and Nigeria joined Algeria, Egypt, Ivory Coast and Ghana in the final elimination round.
The winners of matches between Senegal and Uganda later Saturday and Cameroon and Libya Sunday will complete the 10-nation line-up.
A seeded September 16 draw in Cairo, based on FIFA rankings to be released Thursday, will create five two-leg ties with the winners going to the finals in Brazil.
Ethiopia recovered from conceding a mid-first half goal with Salahdin Ahmed levelling soon after the break and Minyahel Teshome the match-winner.
Success was particularly sweet for Teshome as he cost Ethiopia three points by playing in an away victory over Botswana when suspended.
South Africa dominated neighbours Botswana almost throughout and a late brace from Bernard Parker completed what proved a futile win.
Prejuce Nakoulma was the toast of 2013 Africa Cup runners-up Burkina Faso, scoring nine minutes into the second half.
Congo needed a win in Niger to stay top, but trailed twice to goals from Mahamane Cisse and Daouda Kamilou and had to settle for a 2-2 draw.
Emmanuel Emenike put Nigeria ahead in first-half stoppage time and Victor Moses doubled the lead from a 51st-minute penalty.
Any hope Malawi had of clawing back into the game vanished soon after when Limbikani Mzava was red carded following a second caution.
Ivory Coast and Morocco drew 1-1, Gambia beat Tanzania 2-0, Sierra Leone edged Equatorial Guinea 3-2 and Angola whipped Liberia 4-1 in other games.
South Africa 4 Erasmus 28, Furman 45+2, Parker 84, 90-pen) Botswana 1 (Ramatlhakwane 72)
At Brazzaville, Congo
Central African Republic 1 (Keita 23) Ethiopia 2 (Ahmed 48, Teshome 61)
Match moved from Bangui for security reasons
Note: Ethiopia win group and qualify for third round play-offs
Nigeria 2 (Emenike 45+1, Moses 51-pen) Malawi 0
Note: Nigeria win group
Angola 4 (Guedes 23, Mabululu 48, Guilherme 61, Abdul 64) Liberia 1 (Marcus 79)