The Seagulls had plenty of opportunities to break the deadlock, with Inigo Calderon - replacing the injured Bruno - almost marking his return to the side with an early goal, only to see his shot come back off the post.
Will Hoskins wasted a trio of chances and Brighton were almost made to rue their failure in front of goal when Billy Sharp, who brushed the crossbar, and Dexter Blackstock went close either side of half-time.
Kazenga LuaLua, a 69th-minute change for Hoskins, and Will Buckley almost netted winners late on in a second half starved of chances, but goalkeeper Lee Camp was on top form to deny the duo.
The stalemate leaves Gus Poyet's side without a win in three games now and one spot above ninth-placed Forest in the table.
The hosts almost opened their account in the eighth minute but Calderon saw his left-footed shot from Craig Mackail-Smith's cutback hit the post.
Brighton went close again shortly afterwards when Hoskins shot wide from six yards before Wayne Bridge drilled into the side netting. Poyet's outfit continued to dominate and Hoskins wasted another two opportunities to net the opener, volleying and heading over from crosses.
Camp had to be alert to tip over Calderon's effort in the 39th minute, but the Seagulls almost went behind on the stroke of half-time when Sharp clipped the bar from Cohen's cross.
The visitors started the second half like they finished the first and were close to netting the opening goal as Blackstock poked substitute Greg Halford's cross wide.
Camp was on hand to save Calderon's third effort, while Henri Lansbury - a 59th-minute replacement for Adlene Guedioura - tested Tomasz Kuszczak from 12 yards.
LuaLua almost found the all-important first goal nine minutes from time when he collected the ball on the left and cut inside before unleashing a shot which the diving Camp pushed to safety.
Camp was then on hand to deny Buckley, who had burst through after latching on to a Halford mistake.