Charlton made the perfect start to the match as Tony Watt won an early free kick on the edge of the box to allow Bulot to curl home an unstoppable effort.
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Forest quickly recovered from the early setback to equalise in the 14th minute through Michail Antonio, who cut inside from the left wing to beat two challenges before finding the bottom corner with a powerful low strike.
Dougie Freedman's side grew in confidence after getting back on level terms, and they twice came close to taking the lead as Henri Lansbury tested Stephen Henderson with a curling shot before dragging an effort just wide later in the first half.
Osborn's influence on the game continued to grow, and he came agonisingly close to scoring when his curled effort missed the top corner by a few inches with Henderson beaten.
Charlton survived an extended spell of pressure from the visitors, whose slim hopes of reaching the playoffs were dealt a blow as Dougie Freedman suffered his first defeat since replacing Stuart Pearce.