Pulis, in his second stint at the City helm following a season long exile at Plymouth Argyle, watched as Darel Russell sealed all three points after Pericard had snatched an early lead.
Pericard, making his home debut following a free transfer from Portsmouth, only needed 18 minutes of first-half football to introduce himself to the Britannia faithful.
But the burley frontman had an early shot palmed wide before eventually getting a reward for an industrious opening.
Rams keeper Lee Grant was left asking major questions of his defence when Pericard poked home with some ease after a Peter Sweeney centre had found its way through a packed penalty area.
The hosts should have taken a two-goal cushion into the break when Mamady Sidibe led a lightening break following an over-hit Derby corner.
The energetic striker led the charge from his own box before feeding Sweeney who eventually lost momentum and bent a speculative shot just wide.
City keeper Steve Simonsen was prompted into his acrobatic best just moments later when Ryan Smith let rip a rocket.
At one point Simonsen looked beaten all ends up but the agile stopper somehow managed to get a glove on the former Arsenal starlet's long-range effort.
The second half started at break-neck tempo and it wasn't long before the home side and Russell were reaping the rewards.
Pericard was first unlucky to see his 57th minute strike come back off the bar after latching on to a wonderful reverse pass from Luke Chadwick.
In this mood Stoke didn't need any invitation to double their lead but Grant offered it nonetheless.
The Rams stopper spilled a fierce Pericard drive right at the feet of Russell who duly fired home from close range.
Billy Davies threw on plenty of attacking subs but to no avail as the City defence stood firm Derby were left shooting from range.
Tommy Smith had the best of the late efforts but Steve Simonsen was equal to his drive after the winger got a glimpse of goal.