After falling behind to Heidar Helguson's close-range header in the 11th minute, the Rams, who have lost just once at home this term, enjoyed much the better of proceedings.
Yet it was only after Noone had seen red for two yellow cards in the space of three second-half minutes that Derby found a way through, with Theo Robinson converting a square pass from Craig Bryson.
Both sides had chances prior to the visitors' early opener.
Craig Bellamy should have done better when firing high over the bar from 25 yards, while Derby skipper Richard Keogh leapt highest to meet a right-wing corner, only for Noone to scramble the ball off the line.
Cardiff were soon in front courtesy of Helguson, who was on hand to nod home a rebound after Mark Hudson's looping header had been tipped on to the crossbar by Adam Legzdins.
Michael Jacobs curled an effort narrowly wide and Hudson produced a fine aerial clearance to deny Robinson as Derby responded well.
Nigel Clough's men finished the first half stronger and David Marshall pulled off a fine low save to deny Paul Coutts before Jake Buxton sent a header on to the roof of the net.
Cardiff remained on the back foot after the interval, but could have extended their lead when Bellamy broke clear on the counter. However, the veteran forward once again failed to keep his shot down.
Strong appeals for a Derby penalty were waved away after Noone appeared to block Bryson's goalbound shot with his arm. Yet within seconds, the Cardiff winger was handed a second booking in quick succession following a late challenge on Hughes.
The hosts swiftly capitalised as Bryson surged into the area before squaring to leave Robinson with the simplest of tap-ins.
Helguson almost restored his team's lead when he beat Legzdins to a left-wing cross, yet the momentum remained with Derby until the final-whistle.
Ultimately, the Welsh side were able to hold on, although Paul Connolly was forced to head off the line during six minutes of injury-time.