Magno Vieira gave Barnet some consolation with a goal in the 78th minute as he and Leeds sub Shaun Derry challenged for a corner.
Blackwell began his career at Barnet 21 years ago and the win against his old club silenced calls for the manager's head - at least for the time being.
Leeds goalkeeper Neil Sullivan was fortunate to remain on the field after raising his studs to Vieira in the 51st minute. It looked a definite penalty, but amazingly referee Darren Drysdale and his assistants missed the offence.
Recalled striker Blake struck his first goal since March 4 in the seventh minute, watched by just 7,220 spectators - Leeds' lowest home crowd since 1961.
Among the absentees was chairman Ken Bates, who was back home in Monaco.
Barnet defender Nicky Bailey tried to cut out a long pass from stand-in United captain Gary Kelly, but only succeeded in deflecting it towards Blake, who kept his composure to shoot past Ross Flitney from 19 yards.
Leeds looked sharper after making five changes from the side beaten at Coventry. Barnet, though, played like strangers at times, with eight changes from their home defeat against Notts County.
Bailey almost turned the ball into his own net in trying to make a clearance inside the six-yard box and from the resultant corner Hayden Foxe should have doubled Leeds' advantage, but the Australian headed wide.
Leeds lost Geoff Horsfield with a tight hamstring two minutes before the break, with Moore leaving the bench to partner Blake up front.
Barnet began the second half with Tresor Kandol and Vieira in place of strikers Giuliano Grazioli and Liam Hatch.
Moore's first goal, in the 55th minute, gave Leeds welcome breathing space. Barnet failed to clear Eddie Lewis' corner from the left and Moore headed in from a couple of yards.
His second was drilled in from Blake's pass in the 74th minute and four minutes later Vieira turned Jason Puncheon's corner into the net.
Leeds had the luxury of giving promising teenager Jonathan Howson his debut, the youngster replacing Steve Stone 14 minutes from the end.