The duo struck late goals to defeat Leamington 2-0 in their third qualifying round replay on Tuesday and win a trip to National League North leaders Harrogate Town.
Five minutes from time Shane Clarke was fouled and Jarman curled a 20-yard free kick into the top corner.
Then five minutes into injury-time substitutes Thomas Davie and Wells combined to secure victory moments before the referee blew the final whistle.
It was no more than Trinity deserved after they created the better chances late on in a tight tie.
Seventeen minutes from time Ioan Evans struck the outside of the post and Jarman shot straight at the keeper when well placed.
The replay followed a goalless draw on Saturday when 10-men Trinity drew 0-0 at Leamington, despite a red card for midfielder Jonathan Wafula.
Trinity might have scored in the first minute as Jarman’s shot was blocked after a sweeping move.
The striker again threatened from a Josh Lacey long throw while midway through the opening half Trinity’s loan keeper, Henrik Ravas, denied Thompson-Brown.
Ravas was at his best at the start of the second period to keep out Stefan Moore’s lob.
Those saves proved vital when Trinity, who won 2-1 at Spennymoor Town in the second qualifying round, struck late on for a dramatic win.
The Holy Blues return to league action on Saturday at sixth-placed Tamworth as they look to end a run of three successive defeats in National League North that have left them second from bottom with eight points from 12 matches.