Gainsborough were the only side to defeat a team two divisions or higher than them in the final round of qualifying when they knocked out National League Hartlepool, but found Accrington a much tougher nut to crack from the off.
The home defence had to be alert in the opening stages to keep the game goalless, throwing bodies in the way whenever possible.
They were unlocked just before the half hour mark after a fine move saw Isaac Sinclair brilliantly control a diagonal cross-field path before delivering for Tyler Walton to nod down to Paddy Madden, who converted from close range.
While many in home end felt the striker was ahead of the last defender before the ball was played, the goal stood, and Stanley became the fourth EFL club Madden has scored for in the FA Cup after Stockport, Fleetwood, and Scunthorpe.
From that point on until the break there was only one team in it, but both Walton and Isaac Heath failed to hit the target and extend the visitors’ lead.
A fireworks display briefly provided a noisy backdrop to the start of the second half and seemed to spark the hosts into life.
Frank Mulhern saw two openings slip by. First a diving header into the corner was kept out by Oliver Wright in the Stanley goal, and then the Gainsborough substitute narrowly missed the target with a drive from the edge of the area.
Accrington almost doubled their lead moments later when captain Shaun Whalley curled an effort onto the post, but they were made to pay for their missed opportunities 15 minutes from time when Tuntuwana broke free between defenders from Luke Butroid’s free-kick to nod the hosts levels.
Five minutes into extra-time, however, Caton popped up 10 yards out to steer home the winner from Madden’s pass across the box and end the seventh-tier side’s gallant run in the competition, which started in the first qualifying round.
