Algeria have won the African title once, in 1980, and will be making their first appearance in the semi-final of the tournament since 2010.
Tunisian defender Yassine Meriah told reporters in a post-match interview that his team was more focused, and deserved the win.
Benin and Madagascar's resilient and hubristic run was also halted in the process, though both leave the tournament having won many hearts with their sense of sheer optimism about heading somewhere, when many thought they would be gone earlier.
Tunisia drew four consecutive matches before taking advantage of jaded giantkillers Madagascar to win 3-0 with captain Youssef Msakni among the goals.
"I've known Cisse for a very long time, we used to play against each other when we were both in France. They were different players", recalled Belmadi. "Today we celebrate, tomorrow we're back to business, and we have to concentrate a hundred percent for the next game against Senegal".
- Algeria midfielder Adam Ounas has been involved in a goal every 30 minutes in this year's tournament (three goals, one assist in 121 minutes played), the best ratio in the competition.
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Sophiane Feghouli put Algeria in the lead in the 29th minute, but Baghdad Bounedjah later squandered an opportunity to make it 2-0 by missing a penalty.
Wilfried Bony had his penalty saved in the shootout by goalkeeper Rais Mbolhi to give Algeria the advantage.
After failing to progress past the group stage of the 2017 Africa Cup of Nations, Belmadi has since improved the Desert Foxes and Abeid has lauded the impact of the 43-year-old coach.
"We are not disappointed because we played a big match, then we failed in the penalties", he said.
"Now I see a solid Algeria side with a good balance between attack and defence".
Both teams are led by local coaches in their early 40s with ex-Algeria worldwide Belmadi in charge of the Desert Foxes and Senegal under the leadership of Aliou Cisse, who played for the Lions of Teranga when they last reached the final in 2002.