The Rams delivered a vibrant performance and led when Max Bird celebrated his 21st birthday with a 25-yard strike in the 32nd minute.
Curtis Davies headed a second two minutes later when he beat Josef Bursik to a free-kick and Stoke were relieved to hear the half-time whistle.
But Derby should have wrapped the points up early in the second half when a mix-up between Ben Wilmot and Bursik presented Ravel Morrison with an open goal but he shot into the side netting.
That threatened to be costly when Stoke pulled a goal back in the 58th minute with former Derby winger Tom Ince striding onto a Mario Vrancic pass to fire low past Kelle Roos.
The momentum had shifted and Stoke pressed hard with Ince shooting wide after a corner was flicked on before Wilmot headed just over from another corner.
But Derby’s defence, marshalled impressively by Davies, stood firm to provide some relief from the financial gloom.
Rooney said: “I’ve said a few times I’m committed to this club and to the group of players and the staff, I care about them, so I’ll keep doing everything I can to help us get through this.
“I think we will get through this for the better. We are going to have tough times ahead in the near future but it’s my job now to start rebuilding this club and trying to put it back together.
“My job is to bring some pride and dignity back to the club, it’s got a big fan base and those fans deserve my full attention and the players’ full attention.”
Rooney revealed he only found out about the plan for Derby to go into administration from TV.
“I saw it on the TV which is not ideal, I think communication is key, I’ve still had no contact with Mel (Morris, the owner) for a while now so it’s been very difficult for me to address the fans when I don’t know what the situation is.
“It’s been a very difficult 24 hours finding out the news of going into administration. I imagine there will be people who will lose their jobs who have mortgages and bills to pay so to have that snatched away from them is heartbreaking.
“I’ve got a very good honest group of players, today was a game I obviously wanted to win but I wasn’t that interested in the result, it was more about the players showing character and showing the fans they care.
Stoke manager Michael O’Neill felt errors had cost his side.
He said: “Young players make mistakes and that was what happened today.
“We lose the goal and then we lose the second goal and at that point I thought our reaction was poor but our reaction in the second half was excellent.
“We got a goal back and really should have got ourselves level with the opportunities we had.
“We had two really good chances before we did score but I can’t begrudge Derby the win, they’ve had a really difficult 24 hours.
“I thought they showed great spirit in how they played, the crowd were terrific as well.”