Making his first start in more than three months following injury, Asaba calmly swept the ball home just after the hour.
The win keeps Neil Warnock's side in third place and they are now four points clear of fellow top-six contenders Forest.
After a scrappy opening, Peter Ndlovu looped a header just wide following Michael Brown's free-kick from the left while Marlon Harewood lost control of the ball in a good position at the other end after 11 minutes.
Michael Tonge had a free-kick from 25 yards turned behind by goalkeeper Darren Ward before Brown's tremendous surge down the left ended with a low cross for the advancing Dean Windass, but his shot flew high over the crossbar.
The hosts almost broke the deadlock just after the half hour when Phil Jagielka, operating at the heart of the defence again after impressing against the Gunners, headed the ball against the crossbar following Tonge's corner.
David Johnson turned the ball over after Harewood's lightning burst down the right as Forest continued to look menacing on the break before a defensive mistake gave Ndlovu a clear sight of goal for the Blades, but he fired straight at Ward from 10 yards.
Des Walker was booked for a challenge from behind on Windass before United deservedly went ahead after 61 minutes.
Asaba slotted the ball home from 10 yards after Ndlovu's deflected shot, following a jinking run across the edge of the area, ran into his path.
Johnson lashed a left-foot shot over the crossbar as Forest sought a rapid reply but it was not forthcoming as the bulk of a crowd of 23,317 left Bramall Lane happy.