Weale allowed Paul McKenna's weak shot to squirm through his hands seven minutes from time after Leicester had twice come from behind to level.
After a high-octane start which saw two goals in the space of five first-half minutes from Marcus Tudgay and Matt Oakley, the match appeared to be petering out as players wilted in the scorching spring sunshine.
However Robert Earnshaw pounced to put the hosts back in front with less than 20 minutes left before Foxes substitute Darius Vassell levelled 60 seconds later.
Both sides were guilty of surrendering possession cheaply and a draw looked likely until McKenna struck to earn Forest only their second win in 12 games.
Lewis McGugan hammered a half-volley over the top from Tudgay's chest down in the opening stages before the pair combined for the 15th-minute opener.
McGugan drove in a powerful cross from the right flank and after Tudgay had found a yard of space in the area the former Sheffield Wednesday man steered in a brilliant header to beat Weale.
The goal had been coming but Davies saw his side surrender a deserved lead within five minutes.
Guy Moussi was stripped of possession in midfield and Richie Wellens found Kyle Naughton in space on the right.
His cross was helped across goal by Paul Gallagher and while Moussi hooked the ball clear, Oakley was on hand to rifle in a 25-yard strike which Lee Camp could only touch onto the post and into the back of the net.
It was a full-blooded strike from the veteran midfielder but Camp will feel he should have kept it out.
The sides went in level to the annoyance of the home support behind Davies, while Oakley lasted just five minutes of the second half before he was replaced by Yuki Abe.
The second 45 minutes was struggling to provide a spectacle and Davies responded by readying strikers Kris Boyd and Robbie Findley.
The duo were stood in the technical area ready to come on when Earnshaw struck in the 73rd minute.
Again the dangerous McGugan was involved, twisting away from a challenge to send in a powerful strike from the edge of the box.
Weale failed to hang on to the shot and Earnshaw reacted quickest to slam home.
Again though the hosts failed to hang on and conceded just a minute later with a similar goal.
Naughton was not picked up from a free-kick and fired in a low cross-shot from the right which Camp tipped out and Vassell buried in the roof of the net.
Boyd and Findley were then introduced before McKenna's 84th-minute winner.
Boyd won a challenge but there appeared little danger as the midfielder shinned his 20-yard volley straight at Weale.
However the goalkeeper had a nightmare as he attempted to gather the ball and saw it slip through his grasp and into the net.