Platini opened the scoring for Cape Verde in the 28th minute, before Ramos Nhuck doubled the advantage three minutes before half-time.
Cape Verde held on to their lead throughout the second half, leapfrogging their opponents in the process and advancing to the next stage by virtue of finishing as one of 10 group winners in the CAF section of World Cup qualifying.
Those group winners will be now drawn against each other in five two-legged ties to decide who will travel to Brazil next year, with Burkina Faso also sealing their place in the final round with a 1-0 win over Gabon.
Prejuce Nakoulma scored the only goal nine minutes after the break to send his side through at the expense of Congo, who could only draw 2-2 with Niger.
Mahamane Cisse gave Niger the lead on 34 minutes, with Fabrice Nguessi equalising for Congo midway through the second half.
Daouda Kamilou put Niger in front for a second time on 70 minutes, before Congo levelled through Ulrich Kapolongo with 15 minutes remaining.
Senegal did enough to progress after defeating Group J rivals Uganda 1-0.
A win for Uganda would have seen them qualify ahead of Senegal, but they faced an uphill struggle when Godfrey Walusimbi was shown a straight red card in the 36th minute.
The winner came courtesy of Sadio Mane six minutes from time as Senegal made it safely through.
Ethiopia's passage was less assured as they had to come from behind against Central African Republic to seal their spot in the final round.
Central African Republic went ahead through Salif Keita midway through the first half, but second-half strikes from Saladin Said Ahmed and Menyahel Teshome ensured Ethiopia's progress.
That prospect looked distinctly uncertain at half-time, with South Africa well on their way to the 4-1 win they needed to jump above Ethiopia if had lost.
Nigeria were another side to seal their passage into the final round as they defeated 10-man Malawi 2-0.
A win for Malawi would have seen them qualify, but an Emmanuel Emenike goal in first-half stoppage time and a Victor Moses penalty on 51 minutes saw Nigeria through.
Ivory Coast, already through, drew 1-1 at home with Morocco. Elsewhere in the group, a Mustapha Jarju brace saw Gambia seal a 2-0 home win over Tanzania.
In Group J, Angola thrashed Liberia 4-1 with neither side in with a chance of qualification.
Elsewhere, Sierra Leone ended their campaign with a win as goals from Mustapha Bangura, Ibrahim Kargbo and Alhassan Kamara saw them defeat Equatorial Guinea 3-2 in a dead rubber.
