Reyes allowed Storm to tie at 2-2 before the amiable Filipino cue master, who received a superb reception from the Dutch crowd, showed his true form and ran out one rack after the other.
Reyes faces American Earl Strickland in the quarterfinals.
But while Reyes, who won the recent Gabriels Las Vegas 9-ball Open, breezed through the second stage, compatriot Francisco "Django" Bustamante fell by the wayside, absorbing a 6-8 beating from Nick Van Den Berg in the upper draw of the bracket.