Adel Taarabt pulled a goal back before half-time and the former Tottenham player was a constant thorn in Derby's side as second-half goals from Gavin Mahon, Jay Simpson and a late Akos Buzsaky penalty sealed the points.
Derby took the lead when Savage's quick free-kick caught the Rangers defence napping and Dickov grabbed his first Rams goal courtesy of an emphatic first-time finish.
The Rams almost doubled their advantage after 18 minutes when Gary Teale dispossessed Wayne Routledge and whipped in a cross for Rob Hulse to thunder a header goalwards. Rangers keeper Radek Cerny was equal to the task and produced a stunning reaction save before the rebound evaded the prowling Dickov.
QPR were feeling their way back into the game when Damion Stewart was harshly adjudged to have fouled Dickov on the edge of the penalty area 10 minutes before half-time. From the resulting set-piece, Savage curled a sumptuous 20-yard free-kick into the top corner beyond the helpless Cerny.
In the 40th minute Taarabt proved himself to be more than equal to the standard set by Savage when he was brought down by Paul Connolly 25 yards from goal. The Moroccan playmaker dusted himself down to swerve a clinical low effort around the outside of the Derby wall and into the bottom corner of Stephen Bywater's goal.
The first half ended in a flurry of QPR pressure and less than two minutes after the interval the visitors drew level. Taarabt's teasing ball to the back post was knocked down by Routledge for the onrushing and unmarked Mahon to finish from close range.
Taarabt continued to torment Derby's back line and his clever through-ball released Routledge in the 54th minute, only for the former Tottenham winger to fire straight at Bywater.
The seemingly inevitable happened when QPR took the lead just shy of the hour. Rams defender Shaun Barker failed to make a routine clearance and Simpson strolled through the ensuing panic to finish calmly.
Derby threatened sporadically thereafter, mainly from set-pieces, but Taarabt had the opportunity to put the game beyond doubt when he latched on to substitute Mikele Leigertwood's cross-field ball and bore down on goal to force Bywater into another impressive stop.
Buzsaky settled the issue from the penalty spot in the final minute after Routledge went down under a challenge from Dean Moxey, where contact appeared to be minimal.