Radoslaw Majewski supplied a spectacular 33rd minute opener and two headers in the space of two minutes after the break, starting with a powerful Greg Halford effort in the 65th minute and followed up by another from substitute Matt Derbyshire eased the Reds into the second round.
Neil Austin did pull a goal back with a header of his own in the 76th minute, but Hartlepool, managed by former Forest defender Cooper, rarely looked like troubling the Championship side.
Forest might have been more than a goal to the good by the interval, with Halford having come close with three headers - one of which was cleared off the line - and Scott Flinders having to produce a flurry of smart saves.
After Majewski had broken the deadlock with a fine solo strike from the edge of the box, Gonzalo Jara would have doubled Forest's lead if not for Flinders, who pushed away the right back's powerful drive.
Left back Dan Harding then tested Flinders with a curling shot of his own.
The only disappointment for Forest was seeing Simon Cox carried off with an ankle injury in the 26th minute, after seeming to stand on the ball awkwardly. However, Hartlepool also suffered a blow, with the superb Flinders having to be replaced at half-time due to injury.
Luke Jones had a golden chance to pull the League Two side level, but fluffed his lines by losing his footing at the vital moment.
Then came Forest's quick-fire double, as Halford headed home a Majewski corner and Derbyshire nodded home a fine cross from fellow substitute Andy Reid.
Even after Austin's header from a Darren Holden cross, Hartlepool rarely threatened to claw their way back.