First half goals from Msakni Youssef and Cameroonian Ndjeng Joseph Yannick proved decisive.
Stars coach Gbenga Ogunbote spent much of the pre-match build up talking up the importance of opening with a win.
It took all of 15 minutes for all that to evaporate. Youssef found himself unmarked at the back post as a low cross evaded the badly-positioned Sunshine back line.
The tap in could not have been any simpler.
Sunshine pushed forward in a desperate search to draw level before the break, only to see their adventure punished ruthlessly with literally the last kick of the half.
Yannick let fly with a superb curling effort from distance that Moses Ocheje failed to reach despite straining every sinew in his full length stretch.
Dele Olorundare hit the post in the second half but by then, the Tunisians were coasting, and for the last half hour, they were playing keep ball, to the frustration of their hosts.
Ogunbote still found a way to put a positive spin on the situation.
"We conceded two quick goals because we gave our opponents too much respect. Hopefully, we are going to work on our shortcomings before our next game," he said.
"It’s not over for us and we are looking forward to a better performance in our next match against Etoile."