The visitors took an early two-goal lead when Steve Jones curled the ball past Lee Camp before David Vaughan's free-kick found the top corner.
Marcus Tudgay's flashing header pulled one back and just after half-time Inigo Idiakez lashed home in a crowded area.
Dean Ashton hit the post as Crewe hit back and he headed in after a flowing move to make it 3-2 before Jones took the ball around Camp to seal the win.
Crewe manager Dario Gradi:
"It's only the second time we've played with three forwards. We did it against Blackpool in the week and it worked.
"The three of them are very dangerous players and they are also goal scorers.
"Even at 2-2 I thought we looked like we had more goals in us and Derby had a lot of pressure without really looking like scoring."
Derby: Camp, Doyle, Johnson, Huddlestone, Vincent, Bisgaard,
Taylor, Idiakez, Reich (Junior 60), Smith, Tudgay.
Subs Not Used: Grant, Kenna, Peschisolido, Bolder.
Goals: Tudgay 16, Idiakez 46.
Crewe: Williams, Tonkin, Moses, Foster, Billy Jones, Lunt,
Sorvel, Rivers (Higdon 90), Vaughan, Steve Jones, Ashton.
Subs Not Used: Ince, Walker, Bell, Varney.
Goals: Steve Jones 5, Vaughan 12, Ashton 74, Steve Jones 83.
Ref: L Mason (Lancashire).