The Championship outfit hit a first-half goal followed by two quick-fire strikes after the break to ensure it was their name that went into the draw for the Second Round but Pools did grab a late consolation through Neil Austin.
Pools had some defending to do early on as Forest pushed out from the off to test Cooper’s new-look defence but the young back-line withstood the early pressure to allow some possession at the other end of the pitch.
And, after a Jack Compton freekick into the box, Brad Walker picked up a loose ball on the edge of the area to execute a lovely turn and shot which was held by home stopper Dorus de Vries.
At the other end Jack Baldwin came to the rescue for Pools, blocking an almost-certain goal from Jamie Mackie with a last-gasp stop on the line.
Greg Halford sent a header just wide from a corner kick as the hosts enjoyed the larger share of possession, and Mackie looked set to put his side ahead when he shot from the edge of the box but Flinders denied him with a great catch.
On 33 minutes, the hosts’ pressure paid off as they grabbed the lead through Radoslaw Majewski; Forest put together a neat passing move in the build-up to the goal and a neat piece of skill on the edge of the box bought the goalscorer room to power a drive past Flinders.
Minutes later and the Pools’ keeper was back in action again, this time denying a second with a great save to prevent Gonzalo Jara who spun past Darren Holden to burst in form the right and shoot on goal but Flinders stayed big to deny the Forest full-back.
Simon Walton combined well with Baldwin on the edge of the box to deny another strike as the hosts took the impetus after their opening goal and Pools struggled for a period to get hold of the ball.
Flinders made another save, this time to deny Dan Harding before Pools managed to get the ball up to the right end of the pitch to have their own chance on goal.
Jonathan Franks whipped a decent ball in from the right which Antony Sweeney managed to get a flick on to but unfortunately it was not enough to send it goalwards.
The half ended with another save from Flinders, this time from Mackie, and that was to be the last of the action from the big keeper who was replaced by Andy Rafferty at half time – suffering with an injury.
Rafferty started off the second half much as Flinders had finished the first as he pulled off a steady save to deny Dan Harding with a long-range effort.
After dealing with another bright start from the hosts, Pools settled into their own stride and went on to have their best chance of the game; good work by Neil Austin down the right saw him push upfield and send the ball in for Franks who in turn fed Luke James with just the keeper to beat. But, with the young striker looking set to test de Vries, his feet got caught up and the chance escaped him.
Rafferty pulled off a great save to deny sub Andy Reid from making it 2-0 – the Pools’ stopper tipped a powerful strike over the bar at full stretch. But the resulting corner from Reid went deep into the area where Greg Halford rose highest to send a header into the back of the net to double Forest’s lead despite Austin's attempts to clear on the line.
And Forest, who were looking increasingly composed and in control of this tie now, made it 3-0 just minutes later when Matt Derbyshire sent a looping header past Rafferty and into the back of the net.
Pools did grab deserved consolation when Neil Austin popped up on 75 minutes to get Cooper's men off the mark for the season. A long throw into the area bounced around before the right back took charge and sent a header past de Vries to make it 3-1.
Nialle Rodney made his Pools debut late in the game and set up a chance on goal with his first involvement as he shook off the defender to push into the box and send a teasing ball across goal. Unfortunately the ball narrowly missed both Sweeney and Howard arriving inside the box.
Pools enjoyed some late possession in the game, with both Holden and Franks sending crosses into the box which caused problems, while Franks himself forced a save from de Vries.
Forest were comfortable winners and secured their passage into the Second Round but Colin Cooper will again look to take the positives from this match as Pools prepare for their first home match of the new League Two campaign on Saturday.
Full Time: Nottingham Forest 3 Pools 1
Att: 9081 (198 Pools)
Nottingham Forest: de Vries, Reyes, Halford, Hobbs, Harding, Guedioura, Paterson, Majewski, Cox (Derbyshire 29), Mackie (Reid 61), Henderson (Miller)
Subs Not Used: Darlow, Collins, Cohen, McLaughlin
Pools: Flinders (Rafferty 46), Austin, Baldwin, Burgess, Holden, Walton, Walker, Sweeney, Franks, Compton (Howard 61), James (Rodney 83)
Subs: Collins, Richards, Monkhouse, Poole