It left Paul Hart's promotion chasing side counting the cost of three away defeats in a row and without a win in seven on the road.
Sibon headed Wednesday into the lead a minute before half time and Johnston added the second early in the second half.
Forest could barely produce a scoring chance of note as Wednesday chased down one of the first division's noted sides.
Sibon should have given Wednesday the lead earlier but messed up dramatically after breaking superbly behind the Forest defence in the 12th minute. With only keeper Darren Ward to beat he dragged his shot wide and was left clutching his head in his hands.
Wednesday's bad luck continued as Tony Crane met a corner from Matt Hamshaw with a firm header only for it to smack against the bar.
Johnston demonstrated quick footwork before whipping in a cross that Adam Proudlock, on loan from Wolves, stretched to head over.
The Frenchman was booked for the foul and Steven Haslam's curling free kick put Forest into a panic, allowing Sibon to make up for his earlier miss leaping to head the ball firmly past Ward.
Wednesday had to bring on substitute keeper Chris Stringer in the second half to replace injured veteran Kevin Pressman.
Forest looked to test him early on but a David Johnson header from Jim Brennan's cross was wide and Gareth Williams was high with an optimistic 30 yarder.
It was Wednesday who stunned the visitors when they struck again after 54 minutes. Hamshaw's driven cross from the right curled behind the defence and Johnston stole in to chest the ball over the line.