Joseph had struck twice before the half-hour to cancel Aruba's advantage, but the visitors halved the deficit through Frederic Gomez right on half-time.
With 20 minutes remaining, St. Lucia scored twice in quick succession, with Joesph completing his treble and Kurt Frederick adding a fourth. But Aruba refused to give up and Rensy Barradas squared the tie on aggregate shortly after.
The Preliminary Draw takes place on 30 July in Rio de Janeiro, and St. Lucia will now join Bahamas, Bermuda, US Virgin Islands and the Dominican Republic in the draw. Belize and Montserrat play their second leg on Sunday 17 July to decide the final spot.