Russell Slade's side progressed to the final on a night of high drama that saw them triumph 5-2 over the former European Cup winners, who had held a 2-0 lead from the first leg.
An Arron Davies strike in the 22nd minute had been cancelled out by Forest striker Scott Dobie two minutes after the interval.
But a late own goal from Alan Wright and a 87th minute header from Marcus Stewart sent the tie into extra-time.
Forest found themselves down to ten men as substitute David Prutton picked up a second yellow card.
Then they found themselves behind as Lee Morris made it 4-1 in first minute of extra-time.
Gary Holt quickly pulled Forest level with a close-range finish but, with an injury to Wright effectively reducing the home side to nine men, Davies put Yeovil through as he notched his second of the game in the 109th minute.
Early in the game there seemed to be little danger as Davies picked up possession 40 yards from goal, but the Forest defence backed off nervously and in costly fashion as the winger charged towards the area before striking a blistering low shot that flew beyond Paul Smith and into the bottom corner.
Stewart could have pulled Yeovil level on aggregate with a stinging volley that could have been no more than a foot over.
But the goal also seemed to spark Forest into life and they could have quickly restored their two-goal cushion as Dobie fired inches wide.
Forest needed the agility of Smith to prevent disaster as he produced a remarkable double save to deny firstly Morris and then Stewart as the striker attempted to power home the rebound.
Dobie pulled Forest level two minutes into the second half with a far post header that sent the home fans into raptures.
But the celebrations were premature as Yeovil launched a remarkable late comeback to send the tie into extra-time.
Smith had made another superb double stop to deny Wayne Gray and Morris and also to block another Morris shot as Yeovil built up the pressure.
But he could do little as a third Morris shot bounced back off the foot of the post and was deflected cruelly into the back of the net off Wright to hand the visitors a lifeline.
Still Forest only had a few minutes to hold out, but even the addition of another defender in the form of Wes Morgan did little to help as Stewart pulled Yeovil level with a thumping header.
Things got worse as substitute Prutton was sent off after picking up his second booking in the space of 15 minutes for a lunging, late tackle on Nathan Jones.
And Forest were in nightmare territory as a shocking back pass from Morgan sent Morris clear to put Yeovil ahead on aggregate for the first time in the tie just two minutes into extra-time.
Still the scoring wasn't over as Gary Holt pulled ten-man Forest level within a minute, stroking home a Grant Holt knock-down from close range.
But Davies sealed the tie for Yeovil with his second goal of the game amid scenes of jubilation from the travelling contingent.