Substitute Lewis McGugan slotted home a late winner for the visitors, who defending desperately to withstand some even later Hull pressure to cement their place in the play-off positions.
George Boyd had given the home side the lead on the stroke of half-time, but Darius Henderson equalised shortly after the restart.
The Tigers went ahead in first-half stoppage time when Jack Hobbs headed Robert Koren's corner back into the danger zone and, as the ball bobbled around, Boyd produced a clever finish with his back to goal, flicking the ball over his shoulder and the stranded Karl Darlow.
Forest responded immediately after the restart, Simon Cox's powerful effort from the right channel deflecting narrowly over before a slice of good fortune allowed them to equalise in the 54th minute.
Radoslaw Majewski's cross from the left found Cox six yards out and the former West Brom man volleyed against the bar, but the ball rebounded kindly for Henderson, who made no mistake from 12 yards.
There were just five minutes remaining when Danny Collins' long ball was knocked down by Dexter Blackstock to fellow substitute Billy Sharp, and his drilled effort across the box fell to McGugan, who rolled the ball home from eight yards.
Enterprising wing-back Brady had already seen three free-kicks whistle inches past the frame of the goal, and his 88th-minute set-piece was even closer as it rebounded off the inside of the post before falling to substitute Jay Simpson, whose shot was scrambled off the line by Cohen.
Steve Bruce's men still did not give up, but their fate was sealed when Mohamed Gedo fired over from six yards after Darlow parried Boyd's speculative effort.