But that doesn’t tell half the story of an incredible night, particularly the extra half-hour, which featured six goals.
Twice Shields looked like they might snatch defeat from the jaws of victory.
They conceded an equaliser two minutes into second-half stoppage time, and then two goals in two minutes in the second period of extra-time as a 3-1 lead was wiped out.
Three minutes from the end, though, Briggs struck his second of the game to cap a remarkable tie.
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The Mariners now progress to the second qualifying round, where they will meet Stafford Rangers.
Carl Finnigan had scored a stoppage-time equaliser in the original tie between the teams on Saturday to force a replay.
And the Mariners picked up where they left off last night, with Gavin Cogdon heading in a Matty Pattison cross in the third minute.
A number of opportunities came and went for the hosts as they failed to extend their lead, and Farsley battled their way back in the second half.
They got their reward two minutes into added time as Ryan Watson slammed home from 25 yards to force the tie to a further 30 minutes.
Shields’ response was superb, and two minutes into the extra period, Briggs flicked the ball past his defender before curling an effort into the bottom corner.
Farsley became increasingly frustrated as extra-time progressed, and five minutes into the second period, Andrew Stephenson looked to have wrapped it up as he buried a 20-yard shot past goalkeeper Graeme McKibbin.
There was still life in Farsley, though, and they pulled a goal back when Danny Ellis headed in a left-wing cross.
Shields barely had time to catch their breath before their lead was gone, with Luke Parkin smashing a low effort into the bottom corner from 25 yards out.
At that stage, the momentum was all with the visitors, and they were suddenly favourites to claim a place in the next round.
Shields were determined not to be beaten, though, and there was one last sting in the tail.
A good move led to Pattison getting in behind, and when McKibbin parried his effort, the ball broke the way of Briggs.
He took a touch and then hit a low strike which took a deflection before creeping over the line to spark home celebrations.
It’s certainly never dull at Mariners Park – and this was a thriller to rival any at the ground over the last three seasons.
SOUTH SHIELDS: Connell, Nicholson, Callaghan, Lough, Morse, Arca (c), Smith (Stephenson 74), Briggs, Cogdon (Sullivan 105), Pattison, Finnigan (Storey 74). Subs not used: Norton, Baxter.
Goals: Cogdon (3), Briggs (92, 117), Stephenson (110).