It took 20 spot-kicks at the near-full Nelson Mandela stadium outside Kampala before the visitors emerged 9-8 winners, having lost the second leg 1-0, with recalled Geofrey Massa scoring midway through the opening half.
The final round qualifier finished 1-1 on aggregate, necessitating the shoot-out, in which Zambia converted nine consecutive kicks after captain Christopher Katongo failed to score from the first penalty.
Zambia are no strangers to shoot-outs, defeating fancied Côte d'Ivoire 8-7 on penalties in Libreville last February after 120 goalless minutes to win the trophy for the first time.
Herve Renard, the French coach who guided Zambia to success against the odds, wore the same white shirt and jeans in each match of the 2012 tournament, and in Kampala.
Nigeria progress comfortably
Victor Moses scored twice and Chelsea team-mate John Obi Mikel once as Nigeria crushed Liberia 6-1 in Calabar to advance after the first leg finished 2-2 in Monrovia last month.
Nigeria were favoured to progress, but the winning margin over the Lone Stars exceeded expectations and will cheer coach Stephen Keshi, who promised to quit if his team failed to reach South Africa.
Acquah, from Serie A outfit Parma, was not even in the original squad, but got a late call-up when another Italy-based midfielder, Emmanuel Agyemang Badu, withdrew injured.
Malawi never stopped trying before a crowd that included state president Joyce Banda and troubled goalkeeper Adam Kwarasey occasionally, but more clinical finishing would have given the Black Stars a bigger victory.
Mali book passage in Botswana
Mali shrugged off the absence through injury of experienced midfielders Seydou Keita and Mahamadou Diarra to triumph 4-1 in Botswana after leading by a solitary goal at half-time.