Three first-half strikes sent Steels ahead before another two in the second period, despite a consolation.
Six minutes in, Josh Nodder capitalised on a Hebburn mistake, slotting past U23s' shot-stopper Kieran Hunter to take the lead.
A minute later, Kevin Bolam's side hit the post from a cross, while Nodder shot wide one-on-one at the other end.
Luke Mangham made it two on the 16 mark, executing some brilliant build-up play with Nodder.
Sam Finlaw powered a shot off target from outside the area, before Hebburn's afternoon got even tougher on 26 minutes.
Steels won a header from Ed Hall's deep free-kick, which Ross Goodwin collected and blasted past Hunter from 25-yards.
Mangham almost made it four before the break, but Hunter stayed big and denied him with his leg.
The visitors did grab a fourth three minutes into the second 45, as Hunter's error was seized upon by Goodwin.
Nodder went through on goal in the 57th minute, however Hunter pulled off a fine stop.
Two minutes later though, Mangham dispossessed Jack Donaghy and struck into the bottom left corner for five.
Amar Purewal then went forward, with his close-ranged shot hitting off Hall before going into the net courtesy of Lee Cooksey.
Kurtis Turner forced two good saves from Hunter in the dying stages, the first being a long shot which took a wicked deflection.
Debutant Liam Smith struck a free-kick on the edge with three to go, however the ball sailed narrowly over, with proceedings finishing at 5-1.
Hebburn Town XI: Hunter, Mullen, Donaghy, Devitt, Taylor, Spence, Archer (Ogbewe, 72'), Potter (Smith, 46'), Martin, Purewal (Allasan, 75'), Bell
Unused subs: Aplin
Goals: Cooksey (OG, 62')
Bookings: Archer (36')
Stocksbridge Park Steels XI: Hall, Lemon, Trench, Finlaw (Tinker, 74'), Cooksey, Fielding, Turner, Goodwin, Mangham, Ruthven, Nodder (Lumsden, 81')
Unused subs: McFadyen, Ludlam, Ring
Goals: Nodder (6'), Mangham (16', 59'), Goodwin (26', 48')
Bookings: Turner (78'), Lemmon (79')
MOTM: Damen Mullen
Referee: Sam Parnaby
Attendance: 287