The All Whites extended their unbeaten run in the qualifying campaign to 11 games and will now face the fifth-ranked South American nation in a two-legged clash for a place in Russia.
The visitors went ahead in the 14th minute through Myer Bevan's first international goal after he turned home Kosta Barbarouses' cross at the second attempt.
And it was Barbarouses who created the second when his cross was headed into his own net by Haddis Gagame after 21 minutes .
Micah Leaalafa got the hosts back into the game from the penalty spot in the 28th minute after he picked himself up from being fouled to convert.
The Solomon Islands levelled with 12 minutes remaining when James Naka was brought down in the box allowing Henry Fa'Arodo to stroke home.