Stoke led at half-time thanks to goals from Martin Kolar and Kevin Harper, either side of Dean Ashton's equaliser.
The Canaries then had Darren Huckerby sent off six minutes into the second half for foul and abusive language as he protested over a tackle.
Mamady Sidibe made victory certain for Stoke after 69 minutes to lift Johan Boskamp's team into the top seven.
Norwich boss Nigel Worthington:
"I was unhappy with both goals in the first half. The defending wasn't up to the standard we expect.
"Our play and passing up to then had been excellent. But you can't concede goals like we did in the first half."
Stoke: Simonsen, Buxton, Duberry, Hoefkens, Broomes, Chadwick, Brammer, Junior (Henry 61), Kolar, Harper (Gallagher 79), Sidibe (Dyer 90).
Subs Not Used: de Goey, Taggart.
Booked: Duberry, Hoefkens.
Goals: Kolar 9, Harper 45, Sidibe 69.
Norwich: Green, Colin, Fleming, Shackell, Charlton, Brennan, Marney, Safri, McKenzie (McVeigh 60), Ashton (Doherty 86), Huckerby.
Subs Not Used: Drury, Jarrett, Ward.
Sent Off: Huckerby (52).
Goals: Ashton 38.
Ref: M Pike (Cumbria).