The Whites started brightly and created the first chance of the afternoon in the 11th minute as Charlie Taylor's strike was well saved by Karl Darlow in the visitors' goal.
Forest ventured into the Leeds area moments later as Chris Burke gave his marker the slip and let fly, Giuseppe Bellusci producing a vital block to snuff out his strike.
Osborn caused the hosts more problems six minutes later as he toe-poked Michail Antonio's cross goalwards, forcing Silvestri into action again.
The teams looked inseparable in the first period and clear-cut chances started to dry up as half time approached, but Taylor tested Darlow again in the 33rd minute, his first-time strike needing only a comfortable save.
The game looked equally tight in the early stages of the second half as both sides were restricted to long-range efforts, Taylor coming closest with his on the hour mark when he almost grazed the crossbar from distance.
Forest almost took the lead in the 72nd minute through a spot of good fortune as Eric Lichaj's long-range strike bounced wickedly in front of Silvestri and almost caught out the Italian.
Burke cut into the Leeds area moments later and went for goal, but Silvestri underlined his desire for a clean sheet with another impressive save.