Seasiders boss Curtis Woodhouse made seven changes from the team that started the previous game, against Garforth Town in the FA Cup, for the second round clash at Queensgate.
The Seasiders, who were captained for the night by former Goole midfielder Terry Barwick, looked to be heading into the next round thanks to a 24th-minute goal from Chris Adams, the latter continuing his return from injury.
Strugglers Goole, who were relegated from the Evo-Stik League last season, refused to buckle though and levelled two minutes from time through Ollie Smith, the forward bringing the ball down well before tucking his shot into the bottom corner.
The visitors made the ideal start to the first period of extra-time, with Smith putting them ahead after only a minute.
Goole then secured their place in the next round with a 105th-minute effort from Derry Robson.
The Seasiders will hope to return to winning ways this Saturday when they head to basement club Athersley Rec, who have yet to win a league game this season.
Woodhouse’s troops did the double over Athersley last season, an injury-time goal from Jake Day handing them victory on the road a year ago, then winning 3-1 at home thanks to efforts from Day (2) and Andy Norfolk in March.
Town now know who they will be playing in the FA Vase second qualifying round a week on Saturday, with Consett having claimed a 2-1 victory at North Shields last weekend.
After their defeat against Garforth Town in the FA Cup last month, the Seasiders will be aiming to launch a decent run in the Vase to help boost the coffers. £725 going to the winners of the cup-tie.
Bridlington Town: Jody Barford, Lewis Dennison, Jack Bulless, Jack Griffin, Chris Jenkinson, Fraser Papprill, Chris Adams (Shaq McKenzie 62 mins), Terry Barwick (capt), Nathan Modest, Myles Lawman (Andy Norfolk 62 mins), Ben Flowerdew (Joe McFadyen 62 mins).
Substitutes not used: Anthony Bowsley, Mike Thompson.
Goole: Joe Wilton, Ryan Noon, Adam Lund, Alex Varley, Steve Jeff, Josh Schofield (Max Tweddle 75 minutes), Rob Constable, Derry Robson, Jack Smith, Ollie Smith, Billy Law (Dan Nicholson 75 minutes).
Substitute not used: Marcel Chigumira.
Referee: Samuel Packer (Middlesbrough).