Mahrez struck twice in the opening half hour as Algeria, who won the continental title in 1990, sped into a three-goal lead away against Togo before eventually winning 4-1 to take top place in Group D.
Mauritania qualified for the first time from Group I as they came from a goal down to beat Botswana 2-1 in Nouakchott with Ismael Diakite scoring an 84th minute winner.
Guinea qualified without kicking a ball several hours before their Group H game against the Ivory Coast in Conakry when Rwanda and the Central African Republic drew 2-2 in Kigali.
Guinea then went on to win the group with a 1-1 home draw against the Ivorians, who got the point they needed for qualification when Jean-Michael Seri equalised Mohamed Yattara’s early goal for the hosts.
Burkina Faso, Tanzania and Zimbabwe all missed out on a chance to book their berths. Burkina Faso were beaten 2-1 in Angola while captain Willam Jebor scored the only goal as Liberia checked Zimbabwe 1-0 in Monrovia.
Tanzania lost 1-0 away in Lesotho, who won a rare competitive international with a header from tall centre back Nkau Lerotholi.
The 2012 champions Zambia were eliminated from the race in Group K as they went down in Maputo to Mozambique with Reginaldo's 63rd minute goal separating the two teams.
Jordan Ayew scored twice as he and his brother Dede returned to the Ghana team for the first time in a year to help beat Ethiopia 2-0 away in Group F.
The derby between Congo and the Democratic Republic of Congo ended 1-1 in Brazzaville, keeping both sides’ chances alive in Group G.
The top two teams in the 12 groups qualify except Group B which includes already-qualified tournament hosts Cameroon, where Morocco have made sure of the lone slot.
Egypt, Madagascar, Senegal and Tunisia all booked their place last month while Mali, Morocco, Nigeria and Uganda were added to the list on Saturday.
The rest of the 24-team line-up will be decided in the final round of qualifying in March.
(Reporting by Mark Gleeson; Editing by Christian Radnedge)
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