Goals from Paul Anderson, Dexter Blackstock and Lewis McGugan steered the Reds through.
But, following a 5-0 thrashing at Notts County on Saturday, it was understandable that Bradford's priority was to keep things tight at the back.
And while Forest dominated possession, the Bantams certainly achieved that.
Despite fielding a hugely attacking line-up, the Reds simply couldn't break down Stuart McCall's well organised rearguard during the first 45 minutes.
Bradford's only real effort of the first half saw Lee Bullock power a header narrowly over the bar.
While Forest came close to breaking the deadlock as a Lewis McGugan free-kick, struck accurately from 30-yards, forced a fine save from Simon Eastwood.
And Poland international Radoslaw Majewskii saw an equally venomous long-range strike deflected narrowly around the upright.
Nathan Tyson was brought to a halt by a dubious offside flag as Bradford were broken down on a rare occasion just before half-time.
But with Forest lacking creativity, they deserved to go into the interval with the scores level.
All Bradford's hard work was undone within barely a minute of the second half kicking-off, however, as Tyson stretched his legs down the right flank before delivering an inch-perfect cross that Anderson converted with an emphatic, diving header at the far post.
And on the hour mark, Forest effectively put the game beyond reach as Tyson's pace engineered the room for him to deliver from the left and pick out Blackstock, who forced the ball beyond Eastwood with an acrobatic volley.
McGugan added the third as he slid home an Anderson cross in the 82nd minute.
Bradford were reduced to ten men late in the game when Jonathan Bateson slid in with an awful challenge on Tyson to earn a red card.