Second half goals from Ruben Rochina, Jordan Rhodes and Colin Kazim-Richards settled the contest in Lancashire as Blackburn claimed a second successive win under caretaker boss Gary Bowyer, the son of Forest legend Ian.
It means McLeish will go in search of his first victory as Forest manager at the third attempt in Saturday’s home FA Cup third round tie against Oldham.
McLeish made four changes to the side that had started against Crystal Palace four days earlier with Daniel Ayala, Simon Cox, Henri Lansbury and Guy Moussi returning to the starting line-up.
Radi Majewski, Chris Cohen and Dexter Blackstock were relegated to the bench, while Elliott Ward is back with Norwich after his loan spell ended on Saturday.
Lansbury attempted to make a quick-fire impact on the game but his attempted lob from outside the area after seven minutes failed to hit the target.
Andy Reid followed up his ‘Man of the Match’ display against Palace by picking out Cox six yards out with a superb through ball but the Republic of Ireland international, under pressure from Scott Dann, was unable to get his shot away.
Forest were given a scare when Lee Camp caught what referee Geoff Eltringham deemed to be a back pass from Danny Collins, earning Blackburn an indirect free-kick inside the area.
The decision looked harsh as Collins was attempting to deal with a long punt downfield but the home side were unable to capitalise as Kazim-Richards fired against the 10-man wall protecting Camp’s goal.
Josh King tested Camp with a shot from distance, forcing the Forest No.1 to turn the ball behind for a corner.
Cox had Forest’s first shot on target three minutes before the break but it was easy for Jake Kean to deal with, while Kazim-Richards side-footed on to the roof of the Forest net as Blackburn looked to end the half on a positive note.
The second half was less than three minutes old when Blackburn forged the breakthrough – albeit with a large slice of good fortune.
Rochina made a driving run from midfield and when Ayala came across to tackle him the ball cannoned off the legs of the Blackburn man and beyond the reach of Camp.
It was certainly tough on Forest and they were in danger of falling even further behind moments later as Rhodes fired inches wide of the target after another break from midfield.
The Reds came close to levelling matters just before the hour mark. Harding sent over an inch-perfect cross from the left and although Sharp did everything right with his header Kean somehow managed to get a hand to it and keep it out.
McLeish made his first substitution of the afternoon in the 64th minute when he introduced Cohen at the expense of Lansbury and that was followed nine minutes later by the introduction of a third striker in Blackstock for Guedioura.
But Blackburn extended their lead two minutes later as Rhodes beat Camp from a tight angle after good approach play by substitute Morten Gamst Pedersen and Kazim-Richards.
Things went from bad to worse for The Reds five minutes later as Kazim-Richards made it three by sweeping home a first time shot from the edge of the area after more good work down the right, this time from Adam Henley.
Forest were denied a consolation goal when Blackburn skipper Danny Murphy cleared off the line from Cohen five minutes from time.
Forest: Camp, Halford, Collins(c), Ayala, Harding, Moussi, Lansbury (Cohen 64'), Reid, Guedioura (Blackstock 73'), Cox, Sharp
Subs not used: Evtimov, Gillett, McGugan, Moloney, Majewski
Bookings: Lansbury 34'
Blackburn: Kean, Olsson, Rhodes (Goodwillie 90'), Lowe, Murphy(c), King (Markus Olsson 73'), Dann, Rochina (Pedersen 62'), Henley, Hanley, Kazim-Richards
Subs not used: Usai, Morris, Gomes, Rosado
Scorers: Rochina 48', Rhodes 75', Kazim-Richards 78'
Bookings: Murphy 71', Kazim-Richards 80'
Referee: Geoff Eltringham