Mwila calmed Zambia in the 32nd minute when he completed his brace after Algeria goalie Rais Mbolhi parried Fashion Sakala's ball into his path and the striker made no mistake on the rebound.
Now, they find themselves staring into the abyss after Brian Mwila's double and Enock Mwepu's late strike handed them a 3-1 defeat in Lusaka.
Algeria pulled a goal back with powerful left-footed shot from Yacine Brahimi in the 55th minute and were given a further window of opportunity when debutant Sakala, 20, was dismissed for a second cautionable foul a minute later.
It was the first Zambian win over the Algerians in World Cup qualifying after failing to score in four losses stretching back to the mid-1980s.
The second match-day four group B game holds later on Tuesday when bottom side, Algeria take on Zambia. African champion Cameroon and Ghana are also way off the pace in their groups.
Mwila put Zambia ahead in the sixth minute when he headed in Augustine Mulenga's cross.
A close-range shot from Ghilane Chaalani two minutes into the second half proved decisive and Tunisia finished the match stronger. Both teams need to seriously start picking up points to try and compete with Nigeria in Group B of the competition.
Morocco trounced nine-man Mali 6-0 in Rabat to replace the Ivory Coast as Group C leaders after a match that produced two red and eight yellow cards. Nevertheless, Zambia in its field last Saturday did not leave a chance for the Algerian team, having won with a score of 3: 1.
Hakim Ziyech scored twice for the "Atlas Lions" and Khalid Boutaib, Achraf Hakimi, Faycal Fajr and Mimoun Mahi once. They sit at the fourth spot with just one point.