Swaziland's underdogs had held Nigeria to a goalless draw in the first leg in Lobamba last Friday and it took into the second half of the return game for the west African powerhouse to finally breach their defence.
Ghent forward Simon scored from a free-kick six minutes after the break, before Celtic defender Ambrose turned the ball home from another set-piece with four minutes remaining to seal the tie.
Swaziland finished the game with 10 men when Phumlani Dlamini received his second booking in the closing stages.
Twenty African sides, split into five groups, will compete in the decisive pool stages of the World Cup qualifiers starting late next year, with the top team in each group guaranteed a place in the finals in Russia.
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(Reporting by Nick Said, Editing by Ian Chadband)
