Youssef Msakni beat Super Eagles keeper Maduka Okoye with a long-range effort two minutes into the second half.
Nigeria’s task was made harder when Alex Iwobi was sent off following a video assistant referee review.
The West Africans were the only side to win all three group games in Cameroon, but were largely subdued against a Tunisia side hit by Covid-19.
Nigeria forward Umar Sadiq fired across goal and wide in injury time after Moses Simon had earlier had an effort blocked by Bechir Ben Said.
Tunisia, Nations Cup winners in 2004, will face Burkina Faso in the last eight on Saturday, 29 January.