Rotherham had tremendous backing with more than 2,500 travelling fans certainly making themselves heard in the evenly contested opening exchanges.
Pulis made nine changes from the team which started Saturday's draw at Wigan, but his side had plenty to prove and quickly built up a head of steam.
Stoke should have taken a 14th minute lead when Ryan Shawcross somehow fired Glenn Whelan's free-kick off target from virtually under the crossbar.
And Vincent Pericard faired no better four minutes later when he blasted high and wide of a gaping net from an excellent through ball by Dave Kitson.
But increasing pressure from Stoke gained its reward in the 21st minute after Alex Rhodes was adjugded to have fouled Danny Pugh 20 yards from goal.
Whelan, who has been unable to get into City's starting eleven despite starring for the Republic of Ireland, curled the free-kick over the defensive wall and beyond Andy Warrington's despairing dive.
Stoke continued to pile forward and the visitors were indebted to Warrington in the 32nd minute, when he pulled off a brilliant reflex save to turn Richard Cresswell's shot against a post.
The visitors had failed to test Steve Simonsen in the first half and Stoke continued to make the running, as Pericard fired narrowly wide from 16 yards.
But the underdogs almost levelled in the 53rd minute when Reuben Reid fastened on to a cross from Rhodes to force a tremendous stop from Simonsen with his right hand.
The attempt gave Rotherham some much needed encouragement and Reid was in the thick of the action again, before Drewe Broughton belted a shot over the crossbar.
But Stoke virtually extinguished any chance of a concerted fightback when Danny Pugh doubled their advantage in the 59th minute.
Kitson played a clever ball through and Pugh took the ball in his stride before stroking a left-foot shot inside Warrington's left-hand post from 12 yards.
The Stoke fans then tried to will the hard-working Kitson into his first goal for the club since his £5.5million summer transfer from Reading.
And he thought he had bagged it when he followed in a late effort from Cresswell, until a lineman's flag ruled out his close-range finish.
