Crystal Palace winger Zaha ensured the Ivorians remained unbeaten against Mali at the tournament as he squeezed home on 76 minutes to book a quarter-final showdown with Algeria.
However, the 2015 champions will need to perform much better if they aspire to a second title in four years as Ivory Coast's talented attack of Zaha, Nicolas Pepe and Jonathan Kodjia largely failed to fire.
"We're going to enjoy our victory, it wasn't easy against a good Mali team," said Ivory Coast boss Ibrahim Kamara.
"We had a pretty difficult first half where we couldn't get it going. We let the storm pass and were able to make the most of our key moments."
The in-demand Pepe returned for the Elephants after sitting out the 4-1 win over Namibia along with Kodjia, while Mali recalled a host of regulars having rested key players after qualifying from their group with a game to spare.
Mali advanced as joint top scorers in the group stage with six goals alongside Algeria and had the Ivorians in trouble early here, although Lassana Coulibaly lost his footing after Moussa Marega attempted to square.
Sylvain Gbohouo had a lucky escape when he flapped at a corner with Abdoulay Diaby stabbing over before the Ivory Coast goalkeeper tipped behind his curling free-kick.
Mali were seeking a first victory over their fellow west Africans at the Cup of Nations after three losses and a draw, but Southampton recruit Moussa Djenepo stumbled as he bore in on goal at start of the second half.
Kodjia was denied at the other end as Djigui Diarra turned the Aston Villa forward's shot wide with his legs and an ineffective Pepe was withdrawn on 68 minutes.
Zaha made the breakthrough with a quarter of an hour remaining when Gbohouo's hopeful ball upfield skipped past a static Mali defence all the way through to the Arsenal target who snuck the ball past Diarra.
Adama Traore created a late chance as Mali sought to send the game to extra time but his stinging drive was kept out by Gbohouo.
"We managed the game perfectly and created loads of chances but lacked a finisher today," rued Mali coach Mohamed Magassouba.
"It was a bad day for us. The only chance they had they put it away. I wouldn't even say it was chance, more an opportunist attacker."