Neumann grabbed two of the home side's goals on Thursday, with his first a spectacular effort that gave goalkeeper Phil Moss no chance.
Sergio Cirio, Fabio Ferreira and Awer Mabil were also on target for Adelaide, meaning Kenny Cunningham's fine second-half strike represented nothing more than a consolation for the Phoenix.
To make matters worse for Wellington, Manny Muscat was sent off for a second bookable offence with 17 minutes remaining after fouling the irrepressible Neumann.
Adelaide dominated from the outset and moved ahead in the 23rd minute when Neumann scooted past Michael Boxall and found the net with a stunning volley.
Cirio soon doubled the lead with his fourth goal in two games, having been teed up by a fine cross from Ferreira, who duly made it 3-0 after 59 minutes with an excellent first-time finish.
Ferreira also provided the assist for Jeronimo's second, before Cunningham fired home from range to reduce Wellington's arrears.
Muscat was then dismissed and United added a fifth in injury-time through Mabil after Jeronimo's initial effort had struck the post.
