The leading scorer in the Champions League struck in each half of extra time for the Super Eagles, who will meet the Democratic Republic of Congo on Sunday in a playoff final after they eliminated Cameroon.
The entire playoff bracket is being played in Rabat, Morocco.
The African playoff winner will advance to the six-team intercontinental playoffs in March where two teams will qualify for the the World Cup.
Bolivia (CONMEBOL) and New Caledonia (Oceania) have already qualified for the playoffs.
The four teams in the African playoffs are the best placed runner-ups from nine groups of African qualifying.
In the later match, Democratic Republic of Congo captain Chancel Mbemba snuck in at the back post to score a dramatic last-gasp winner for his team as they edged Cameroon 1-0.
Unmarked Lille star Mbemba forced home a corner from Brian Cipenga to secure a victory for his country and keep them on course for possible qualification to next year's finals.
Group winners Algeria, Cape Verde, Côte d'Ivoire, Egypt, Ghana, Morocco, Senegal, South Africa and Tunisia have already qualified for the tournament to be co-hosted by the United States, Canada and Mexico.
Nigeria last qualified for a World Cup tournament in 2018.
Their clash finished 1-1 after 90 minutes. Akor Adams scored late for Nigeria before Mario Lemina equalised with a deflected shot in the final minute of regulation.
Substitute Chidera Ejuke restored a one-goal advantage for the Super Eagles before Osimhen took charge.
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