The first leg of the playoff finished in a goalless draw at Kumasi but Ghana bagged a crucial away goal through Thomas Partey inside 10 minutes.
Nigeria equalised after 22 minutes through a William Troost-Ekong penalty but the Black Stars held on and booked a place in Qatar later this year.
It will be Ghana's first appearance at the tournament since 2014 after missing out on the World Cup in Russia in 2018.
Nigeria had qualified for six of the previous seven World Cups dating back to the United States in 1994, and their absence from this year's tournament will spark an inquest from the country's football authorities.
But for Ghana, and their newly appointed technical advisor Chris Hughton, it is a major triumph in a qualification campaign in which they never really impressed and had looked a shadow of the side that reached the World Cup quarterfinals in 2010.
They defended for much of the 90 minutes in Abuja, rarely looking threatening after they scored, but also faced few alarms themselves from a Nigeria side devoid of attacking threat.
Nigeria thought they had a second when Victor Osimhen scored, but the VAR correctly ruled he was offside in the build-up.
Emmanuel Dennis headed wide for the home side in the second half and Osimhen sent a bicycle-kick past the post, but it was largely comfortable for Ghana in the closing stages as they held on to the dismay of most of the 60,000 supporters in Abuja.
Information from Reuters was included in this report.