Ghana have nine points at the head of Group H and could secure a place at next year's finals in Gabon as early as Sunday when they play Mozambique again, this time in Maputo.
Anderlecht striker Acheampong ghosted past the visiting defence on the left wing before finishing Mubarak Wakaso's pass to open the scoring in the fifth minute.
Captain John Boye added a second from a corner in the 57th minute, although he appeared to handle the ball in the process, and Aston Villa striker Jordan Ayew chipped goalkeeper Soarito from the edge of the penalty a minute later to seal the win.
Mozambique, playing their first match under new coach Abel Xavier, got a consolation goal through Sonito but have lost all three qualifiers and have almost no chance of making next January's finals.
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Tiny Comoros Islands won a first ever qualifying match as they beat Botswana 1-0 in Moroni earlier on Thursday with Greek-based striker El Fardou Ben Nabouhane scoring the only goal.
The Comoros are participating in only their third Nations Cup qualifying competition and failed to register a win in 10 previous matches.
(Reporting by Mark Gleeson in Cape Town; Editing by Martyn Herman; mark.gleeson@thomsonreuters.com; +27828257807; Reuters Messaging: Reuters Messaging: mark.gleeson.thomsonreuters.com@reuters.net)
