Former Kaizer Chiefs striker Knowledge Musona was the star of the show, bagging a hat-trick on the day with three well-taken goals to propel his side to victory.
The Belgium-based forward was also captaining his side for the first time, having taken the armband from midfield enforcer Willard Katsande.
The victory puts the Warriors top of Group G via goal difference, with DR Congo in second after their 3-1 win over Congo.
Elsewhere, Ethiopia were no match for hosts Ghana in Group F as the Black Stars romped to a 5-0 win thanks to a Raphael Dwamena brace, with Ebenzeer Ofori, John Boye and Asamoha Gyan also on the scoresheet. Ghana are top of Group F alongside Sierra Leone, who were surprise 2-1 winners over Kenya.
Central African Republic join Guinea at the summit of Group H after their 2-1 win over Rwanda, while Benin went top of Group D with a 1-0 win over Gambia.
The 12 winners of each group, along with the best three group runners-up, will qualify for the 2019 Afcon, along with hosts Cameroon.
Sunday’s Afcon qualifying results:
Zimbabwe 3-0 Liberia
Central African Republic 2-1 Rwanda
Benin 1-0 Gambia
Ghana 5-0 Ethiopia