Derby advanced to the League Cup semi-finals after Nathan Ellington's last-gasp penalty sealed a dramatic 1-0 win at Stoke on Tuesday.
Ellington converted from the spot in stoppage time at the Britannia Stadium after referee Rob Styles gave handball against Andy Griffin.
Stoke have turned their stadium into something of a fortress since being promoted to the Premier League, with Arsenal and Tottenham both beaten in the Potteries, but Paul Jewell's Championship club were good value for their shock quarter-final victory.
Derby threatened first when Paul Connolly's cross was cleared back to him and the defender fired in a low shot that Steve Simonsen initially spilled, then clutched at the second attempt.
Pulis's side thought they had taken the lead after 36 minutes when Richard Cresswell drilled a low shot past Roy Carroll, but the striker's effort was ruled out for hand-ball.
While Rory Delap's devastating long throws for Stoke have become a Premier League talking point, Derby weren't going to be intimidated.
They defended Delap's throws well and produced one of their own after 41 minutes. Connolly's howitzer was headed out to Przemyslaw Kazmierczak and he drove just over the bar.
As snow flurries descended on the Britannia Stadium early in the second half, Derby tried to raise the temperature and Kris Commons tested Simonsen within minutes.
But Stoke should have been in the lead after 54 minutes when Ricardo Fuller broke clear of the offside trap, only for Jordan Stewart to frustrate the Jamaican with a superb tackle.
Fuller then headed well wide from close range before Derby midfielder Commons spun on the edge of the box and shot narrowly wide.
Derby went agonisingly close to the opener when Miles Addison hooked against the foot of a post from a 76th-minute corner.
Darren Powell was booked for a foul on Fuller in the closing stages and from the free-kick Glenn Whelan curled in a fine effort that Carroll superbly tipped over.
With extra-time just seconds away, Derby snatched the winner in stoppage time when Griffin was judged to have handled a cross from Kazmierczak.
Styles awarded the spot-kick and Ellington stepped up to send Derby through with a cool finish.
