Esperance now lead their group after the first round of games after local rivals Etoile du Sahel also won 1-0 at Aso Chlef of Algeria on Saturday.
In the second round of matches to be played between 20 and 21 July, Esperance will be home to Aso Chlef of Algeria, while Etoile welcome Sunshine Stars.
The perennial Tunisian champions said they took their chances to win in Nigeria. "It was a good game, we did our best and took our chances," said Harrison Afful. "The victory has put us in good mood to excel in the group. We will take the games as they come."
Goalkeeper Moez Ben Cherifia said they hope to defend the trophy they won last year. "We came here to play and win and that we did. We want to retain the trophy and this is a good step towards that," he declared.
Esperance dominated the first 45 minutes and scored two well-taken goals. However, after the interval Sunshine improved only to be denied by the brilliance of Esperance goalkeeper Ben Cherifia and the post.
Esperance went in front in the 20th minute through Tunisian star Youssef Msakni, who tucked away a superb low cross from the right by the lively Ghanaian international Afful. Sunshine would have drawn level almost immediately, but Osas Idehen's effort from inside the Esperance goal area came off the post.
The Tunisians doubled their lead in first-half injury time when big Cameroonian forward Yannick Ndjeng controlled the ball at the top of the area before firing beyond the diving goalkeeper Moses Ocheje. Before then, Ocheje had to make a reflex save in the 39th minute to keep out the Tunisians.
In the second half, Sunshine fought back in search of goals, but Esperance goalkeeper Ben Cherifia pulled off big saves in the 81st and 87th minutes. Sunshine substitute Dele Olorundare was denied by the post on 67 minutes with Ben Cherifa well beaten.