Sliti broke the stubborn resistance of the Equatoguineans on 56 minutes when, after two failed attempts by the visitors to clear a cross, the Saudi Arabia-based forward slammed the ball into the net.
After Joan Elo was sent off following a second yellow card, a Sliti cross enabled substitute Seifeddine Jaziri to ned past goalkeeper Jesus Owono on 77 minutes at Rades on the outskirts of Tunis.
Another substitute, veteran Youssef Msakni, had a dramatic impact by scoring with his feet on 80 and 85 minutes, and a Sliti pass set up the first.
Next for the Carthage Eagles is a visit to Botswana for another Cup of Nations Group J fixture on Sunday after travelling virtually the length of the vast continent.
Tunisia then face an even longer journey, to Japan, for Kirin Cup matches against Chile on June 10 and Ghana or Japan four days later.
These matches and two Cup of Nations clashes during September with neighbours Libya, who beat Botswana 1-0 on Wednesday, will help prepare for a difficult World Cup mission in November.
The six-time World Cup qualifiers have been drawn in Group D with title-holders France, Denmark and Australia, United Arab Emirates or Peru, and will be the underdogs.
Tunisia have proved good at qualifying for the global football showcase, but poor competitors, winning just two of 15 group matches in five appearances.
– Mozambique disappoint –
In Johannesburg, Mozambique made a disappointing start in Group L, drawing 1-1 with visiting Rwanda in a match moved to the South African city because there is no international-standard stadium in Maputo.
Theoretically, this was the easiest of the six fixtures the Mozambican Mambas (Snakes) will play and recently hired coach Chiquinho Conde showed his frustration when the final whistle blew.
Mozambique had twice as much of the ball as Rwanda, had more goal attempts and shots on target and passed more accurately, but lacked a clinical forward.
Rwanda were playing their first competitive match under Spaniard Carlos Alos, whose coaching CV includes spells with clubs in Hong Kong, Poland, Kazakhstan, Morocco, Qatar and Cyprus.
The Wasps from east Africa snatched a surprise lead on 65 minutes when a Blaise Nishimwe lob punished out-of-position goalkeeper Hernani Siluane and landed in the net.
Rattled Mozambique levelled within two minutes thanks to a glancing header from Stanley Ratifo, who plays in the German fifth division.
Defending champions Senegal host Benin in the same group on Saturday with the home team having to play behind closed doors after crowd trouble during a recent World Cup play-off against Egypt.
There are four Cup of Nations qualifiers on Friday with the Group H clash of the Ivory Coast and Zambia in Yamoussoukro the pick. It is a repeat of the 2012 final, surprisingly won by the Zambians.