Striker Akor Adams snapped up a shallow back pass from Gabon defender Aaron Appindangoye and caught goalkeeper Loyce Mbaba way out of his goal for an easy tap-in for Nigeria’s opening goal in the 78th minute.
However, Mario Lemina then snatched an 89th-minute equaliser, helped by a big deflection, to take the game into an additional 30 minutes of extra time.
A goal from Chidera Ejuke and two from Victor Osimhen then made sure of Nigeria's progress to the playoff final in the Moroccan capital on Sunday, where they will next take on either Cameroon or the Democratic Republic of Congo.
The playoffs are for the four best runners-up across the nine African qualifying groups that concluded last month, with the winners all advancing to next year’s finals in Canada, Mexico and the United States.
The winner of the playoff in Morocco proceeds to represent Africa in March's inter-confederation playoffs, where the last two places for the expanded 48-team World Cup will be decided.
It was an energy-sapping contest in the rain, with Nigeria fortunate not to concede a penalty early in the second half after Bright Osayi-Samuel had looked to pull the shirt of Appindangoye at a corner and prevent him from getting to the ball. But it was not given even after a lengthy VAR check.
Adams, winning only his third cap, then opened the scoring, but Lemina’s strike took a deflection off Osayi-Samuel to hand Gabon an unlikely lifeline.
Osimhen goes from zero to hero
Nigeria might yet have won the game in the 12 minutes of stoppage time that were added at the end of the second half but Osimhen, with only the goalkeeper to beat, inexplicably struck his shot well wide.
However, captain Wilfred Ndidi's superb pass set up Ejuke seven minutes into extra time to score from the right flank and restore Nigeria's lead.
Osimhen then made up for the horror miss with a superb goal five minutes later, netting from a tight angle after being set up by Benjamin Fredrick's storming upfield run.
The former African Footballer of the Year then showed his craft as he chested down a long ball on the edge of the Gabon penalty area, powered around the defence and shot home from the right across the goalkeeper and into the left-hand corner of the goal.
Nigeria's preparations earlier this week were curtailed as their players refused to train until money promised them previously was paid, a situation resolved on Wednesday.