Noone raced to the stands to enjoy his 10th-minute goal with the former Trotters man, who retired earlier this month with a Motor Neurone Disease diagnosis.
Frank Lampard's Rams, who beat Manchester United in the mid-week, could not break down a stubborn Wanderers as they drew a blank in the Championship clash.
Darby, 29, appeared elated with the gesture as he hugged former team-mate Noone, 30.
His wife Steph Houghton has also taken inspiration from her hubby's fight with MND after she scored a worldie equaliser for Manchester City women in the week.
Meanwhile Mason Mount was ill before kick-off for the visiting Rams, as the Chelsea loanee still started the game.
Lampard admitted the bug did affect Mount's game.
The Derby boss said: "Mason felt well enough to play and with Mason you feel he will make something happen.
"He managed to get through, but maybe he wasn’t quite at his best because of that.
"It is very frustrating to lose after good performances in the last two or three games.
"Once you go behind at these places, it’s tough. It is just one of those days.
"The beauty of the Championship is that you do move on quickly."
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