Leading 2-1 from the first leg on Saturday, Blackpool fell behind on the night at the City Groundto Rob Earnshaw's early goal.
Campbell equalised at the start of the second half before Earnshaw scored his second goal of the game to level up the tie on aggregate at 3-3.
Substitute Stephen Dobbie then made an immediate impact with his deflected shot wrong-footing Lee Camp and trickling into the net before two quickfire goals from Campbell sent Blackpool to the Wembley showpiece on May 22.
Forest substitute Dele Adebola scored in injury time to make it 4-3 on the night but it was mere consolation for the hosts.
Forest made one change from the first leg with leading scorer Earnshaw recalled at the expense of midfielder Paul Anderson, who dropped down to the bench.
Blackpool boss Ian Holloway named the same starting XI.
The first chance came after six minutes. Gary Taylor-Fletcher chested the ball into the path of Brett Ormerod whose 25-yard volley dipped just over the crossbar.
A minute later Forest took the lead and levelled up the tie on aggregate. Nottingham-born defender Alex Baptiste slipped trying to intercept Radoslaw Majewksi's reverse pass and Earnshaw fired a left-foot shot across Matt Gilks and into the far corner.
Earnshaw threatened again soon afterwards but Baptiste made amends with a last-ditch sliding tackle to thwart the Wales international.
A fine fingertip save from Camp to flick Charlie Adam's long-range shot over the crossbar midway through the first half preserved Forest's lead on the night.
Chris Cohen came close to making it 2-0 when his low shot from the edge of the area had Gilks scrambling across his goal but the ball rolled just wide of the post.
Dexter Blackstock failed to provide the finish a flowing Forest move deserved on the half-hour.
Some fine one-touch football involving Cohen, Earnshaw and Blackstock saw Forest advance to the edge of the Blackpool box but Blackstock's shot was weak and straight at Gilks.
Earnshaw thought he had scored his and Forest's second goal of the night a minute after the restart.
The diminutive striker lifted the ball over the advancing Gilks after being put through by Blackstock but the assistant referee, somewhat belatedly, raised his flag for offside.
Blackpool equalised and regained the advantage after 56 minutes. Taylor-Fletcher's angled through ball released Campbell down the right side of the box and the on-loan Leicester striker deftly dinked the ball over the advancing Camp.
Earnshaw blasted the ball into the roof of the net from eight yards after 66 minutes to level up the tie at 3-3.
Dobbie, a 69th-minute substitute, made it 3-2 to Blackpool on the night when his shot from the edge of the area went in off Wes Morgan after 72 minutes.
Gilks denied Blackstock with his feet before Earnshaw's goalbound follow-up was blocked inside the six-yard box by a Blackpool defender.
Dobbie played a one-two with Campbell and he finished calmly to put Blackpool in command with 13 minutes remaining.
Campbell then completed his hat-trick soon afterwards after Camp could only push Dobbie's shot into his path.
Campbell struck the bar in the 90th minute before Adebola tapped home from close range to complete the scoring.