Graham Ward’s Meadowmen were heading out of the competition at the first hurdle after finding themselves 3-1 down with nine minutes left against league rivals Wednesfield, but forced spot-kicks deep into stoppage time with a last-gasp Jason Pike penalty.
Glynn Coney and Cameron Milne were also on target for Bridgnorth, who prevailed on penalties.
Ward said: “It was a rollercoaster. We gave away soft goals.
“At 3-1 down we changed shape and went for it. The lads’ belief and courage got them back in the game and we missed a chance to win it 4-3. But we finished the job on penalties. All credit to the lads.”
Bridgnorth go to Pershore Town in the second qualifying round on October 10 after the Worcestershire outfit cruised past St Martins at Park Hall 4-1.
Joining them in the next round are Connor Courtney and Matt Burton’s Ellesmere Rangers.
Dave Easthope, Karl Bailey and Nick Porter were on target for Rangers, who go to Rugby Town next.
Wem Town crashed out 3-0 at Dudley Sports. It was also the end of Whitchurch Alport’s Vase campaign after a 2-0 home defeat to Littleton.
Midland League side Haughmond suffered penalty shoot-out heartbreak at Leicester Road, missing all their spot-kicks in a 3-0 defeat after a goalless draw.
Shawbury United paid for a first-half no-show to exit at Abbey Hulton. Declan Allen’s side were 3-0 down at the break and second-half goals from Reiss Corfield and debutant Seth Ellis were not quite enough.
In league action, Shifnal Town got their West Midlands Premier Division campaign off to a flying start with a 5-1 win at Smethwick Rangers.
Kyle Montague’s brace was added to with goals from Jay Holdcroft, Tom Hill and Dave Howarth for Danny Carter and Andrew Carrier’s men.