Both sides traded early chances before Fanne curled an effort just wide. Moments later, he found the net with a similar chance from the edge of the area after being fed by Wells, despite being off balance.
Wells pounced to double the lead on the half-hour mark after Fiachra Pagel spilled Cohen Bramall's free-kick into his path. The striker should have had another goal a second a minute later but he fired over after springing the offside trap.
Jili Buyabu thought he had halved the deficit with half-time looming. A well-worked corner kick found him in space but his attempt at goal was blocked by Teden Mengi.
Wells made it 3-0 with another close-range finish as Fanne repaid the compliment from his earlier assist after Tre Pemberton was dispossessed high up the pitch.
On the hour, Knowles pulled Rovers back into the game with a curling effort from the edge of the Luton box, which wrongfooted keeper Josh Keeley after a deflection off Mengi, but few in the ground could have foreseen what was to come.
Mengi's night took a turn for the worse just seven minutes later when he brought down Harry Whitwell in the area to concede a penalty kick, and McAllister stepped up to make it 3-2.
With 11 minutes remaining, the ball fell to debutant Dausch after Luton failed to clear and he levelled the scores, sending Forest Green boss Robbie Savage and the travelling fans wild.
Rovers captain Laurent Mendy almost completed an amazing turnaround but his header hit the crossbar with four minutes to go.
Kodua broke Rovers hearts in stoppage time when he smashed home from close range, but there was still time for Dausch to squander a wonderful chance to take the tie into extra time.