Leading 2-1 from the first leg on Saturday, Blackpool fell behind on the night to 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 had taken the lead and levelled up the tie on aggregate after seven minutes. 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.
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.
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.
Dele Adebola grabbed an injury time consolation for the hosts.
FOREST: Camp, Perch (McGoldrick 82), Morgan, Wilson, Gunter, Majewski, McKenna (Anderson 78), Cohen, Tyson, Blackstock (Adebola 78), Earnshaw.
BLACKPOOL: Gilks, Coleman, Evatt, Baptiste, Crainey, Taylor-Fletcher (Burgess 77), Southern, Adam (Edwards 90), Vaughan, Campbell, Ormerod (Dobbie 69).
Referee: Mark Clattenburg.