Tom Ince had a couple of efforts just ahead of the half-hour mark, but both were blocked, while Stockdale pulled off a save in first-half stoppage time to deny the on-loan winger again.
Hendrick came close again when Craig Forsyth curled the ball in from a free kick, leaving the midfielder to attempt to slice it in from the edge of the box, but it was taken by Stockdale.
Brighton weathered the storm though, and started to look dangerous after the hour mark as Dunk came close with a header from a corner, nodding the ball just wide.
Chris Hughton's side broke the deadlock just a few minutes later, when Derby failed to clear a Beram Kayal ball into the box, allowing Dale Stephens, who had just come off the bench, to pounce on the loose ball and fire it home.
Jesse Lingard had a chance to get Derby back on level terms just five minutes later, but he fired over after hitting it first time from Ince's delivery.
Ince and Simon Dawkins both created late chances for the visitors as they tried to give themselves some late hope, but Stockdale made sure that the hosts kept a clean sheet.
With Middlesbrough winning this evening, Derby slip to second place in the table, while Brighton, who have now won three of their last four, jump up to 18th.