United's cause was helped when Forest had Yohan Benalouane sent off two minutes into the second half.
United went into the game on the back of just one win in their last four matches, desperately needing a victory to boost their chances of securing automatic promotion.
Oliver Norwood tried an early effort for the home side, sending a long-range shot wide of the target.
Claudio Yacob had the first sight of goal for visitors Forest when he saw his fierce drive miss the target from 25 yards.
Forest's Jack Colback was then shown a yellow card for simulation when he went down in the area following a challenge from George Baldock.
The hosts had a decent spell of possession, with Stevens' delivery from the left posing problems for the opposition.
First, Stevens whipped in a low, in-swinging cross, which just evaded Gary Madine.
The wing-back then delivered a cross which was headed beyond the far post by David McGoldrick.
Another cross from Stevens again picked out McGoldrick, whose knock-down failed to find Madine.
Forest responded with Daryl Murphy running at the defence before shooting narrowly wide.
Shortly before the break, Norwood tried an effort from long range which was saved by Costel Pantilimon.
Yacob then had to hack the ball clear from inside the area when Stevens threatened to find a team-mate in a dangerous position.
The second period was less than two minutes old when Forest were reduced to 10 men. Benalouane was sent off after fouling McGoldrick just outside the area as the striker threatened to burst through.
From the resulting free-kick, the ball was laid off to John Fleck whose low drive was straight at Pantilimon.
Duffy struck the opening goal in the 51st minute, steadying himself in the left side of the area before arrowing his shot beyond Pantilimon into the top left corner of the keeper's net.
The Forest goalkeeper made a fine reaction save moments later to keep out McGoldrick's close-range effort.
As United continued to apply pressure, Norwood put an effort wide following a corner from Fleck, and Pantilimon was forced to touch a Duffy cross over the bar.
Forest never seriously threatened to mount a comeback, failing to test keeper Dean Henderson.
Stevens added a second goal in the 85th minute to make it a comfortable end to the game for the hosts.
McGoldrick received the ball just inside Forest's half and advanced towards the box before slipping the ball to Stevens on the left side of the area and he produced a low, angled shot to beat Pantilimon.