Smith had another headed attempt midway through the first half, but could only watch as De Vries got down to save the low effort.
Fulham continued to put pressure on the Forest defence, and came close to going ahead through an own goal when Ben Pringle's cross was diverted just beyond the post by Mills.
Despite having much pressure thrown at them during the half, it was Forest who took the lead just before the break when Mills headed in from close range.
Tom Cairney attempted to cut Forest's lead in half with just under half-an-hour remaining, but his effort fired the wrong side of the post.
The Nottingham-born midfielder had another chance with 15 minutes left but failed to properly connect with McCormack's cross, heading wide from six yards out.
Forest took full advantage of Fulham's missed opportunities, and secured all three points when Mills grabbed his second of the game with 12 minutes remaining, connecting with David Vaughan's corner to head in.
The result moves Forest above their opponents and into 15th, with Fulham dropping three places to 16th.