Despite the slender margin of victory, it was an entertaining, open encounter with chances at either end.
But Reid's strike - his sixth goal in the last five games - provided Blues with an outside chance of qualifying for the knockout stages of the competition.
They moved level with second place Wolves - Derby top the group - but the Molineux men have to lose their final two games for Blues to progress.
Marcel Oakley made an impressive debut at right-back, four months past his 16th birthday.
And captain Ryan Stirk was excellent in midfield, getting through a tremendous amount of work and always being there to close gaps and keep play moving.
Early on Reid's movement caused an issue and Jack Concannon was bumped off the ball just as he shaped to shoot from inside the area from his pass.
And that was the way the first-half went with Blues continually finding space and getting forward quickly to create promising situations.
Derby, too, were more than handy when they raided and they fashioned a big chance in the 21st minute for Josh Shonibare, but he dragged a first time shot wide from the left of the area.
A minute later it was Blues turn to threaten when Reid broke away down the left, angled across into the area bypassing three challenges only to blast wide of the target from a great position, after delaying the final shot.
Reid, though, did have more joy in the 24th minute, putting Blues ahead after a left wing corner was cleared.
Oakley floated the ball back in to the far post from the right, Josh Andrews got up and nodded down and Reid took a quick touch before hammering in from 15 yards.
Blues had to be on their toes soon after following a cross from the left. Moha Ramos made a good save from Morgan Whittaker and then Nico Gordon produced a tremendous block from Cameron Cresswell's follow-up.
The game flowed back and forth with Reid then forcing a save out of the 'keeper from the edge of the area after Concannon fizzed a lovely pass into his feet.
In the 37th minute a left wing centre by Kyle Hust was just missed by Andrews. Kane Thompson-Sommers retrieved the ball and wriggled his way towards goal from a tight angle and brought out a near post stop by Matt Yates.
As Derby scrambled to end the danger, they conceded a free-kick a few yards outside the box, which Stirk took. He arched a fierce shot towards the top corner which was superbly tipped over by Yates.
Just before the break it was Blues turn to repel their opponents due to some brilliant last-ditch defending.
Mitch Robers slid across to block George Evans in the heart of the area and when the loose ball was fired back in from Jamal Hector-Ingram, Ramos made a fine save despite seeing the attempt swerve through a crowd of bodies late. The danger was still not over as the ball bounced to Cresswell whose first swipe at it was miscued, which gave Ryan Burke that extra split second to fling himself at the powerful second effort and take its full force.
Blues again had a good break at the beginning of the second period, Thompson-Sommers surging forward on a three-on-three, only for him to be denied as he delayed a pass just a fraction.
Blues had Derby living on edge but at the other end the defensive line also had to show resilience.
In the 66th minute, amid pressure from a corner, Cresswell banged a low shot into a clutch of players amassed in the box and it was heading into the bottom corner before Stirk intervened.
Three minutes later the alarm was back on Derby, from Hurst's cross from the left flank. Concannon prodded a shot that reared off Alex Babos' thigh and seemed destined to float in, but Yates again was equal to things, adjusting to claw the effort over.
A sweeping move in the 75th minute, which went across field from Stirk, to Gordon, to substitute Tate Campbell and involved a combination between Oakley and Concannon, presented Reid with the ball in a good position and he veered across onto his left foot and whacked a fierce effort that whistled just past the bar.
It would have been a glorious way to have killed the group fixture entirely.
But as it was Blues thereafter kept their focus and came through without further scares in what had been a rollercoaster of a match.
Blues: Ramos, Oakley, Burke, Stirk, Roberts, Gordon, Thompson-Sommers (Campbell 68), Concannon, Reid, Andrews, Hurst (George 80). Subs not used: Forrest, Bradley-Hurst, Hilton.