Savage took the gloves in the 41st minute when substitute Saul Deeney was red carded on his Rams debut and pulled off a stunning second-half save before being beaten.
Deeney had come on less than half an hour earlier when Stephen Bywater was injured to leave manager Nigel Clough with the unenviable necessity of sending Savage between the posts.
On-song striker Shane Long opened the scoring for the hosts but Clough's side rallied when Gilles Sunu scored his first professional goal.
Jimmy Kebe had Reading back ahead before Deeney was dismissed and Long struck the resulting penalty high over the crossbar.
Savage then entertained the crowd with a couple of fine stops before Ryan Bertrand hit a deflected shot past him to kill the game off and Simon Church grabbed a late fourth.
The win lifts Brian McDermott's men points four clear of the drop zone while Derby remain in trouble themselves.
The in-form Kebe flashed a cross across the face of goal before he created the opener for Long in the eighth minute.
Former Derby favourite Grzegorz Rasiak started the move with a fine turn before spreading the ball wide to Kebe.
Kebe drilled a first-time pass into Long's feet but the striker showed great to touch to kill the ball and then knock it past defender Shaun Barker.
There was no contest in the sprint for the ball and Long confidently struck his shot past Bywater and into the far corner for his eighth of the season.
Bywater then hobbled off with an injury to give replacement Deeney his Rams debut before Kebe scooped an effort over after Rasiak had seen a volley blocked.
Clough's side had failed to show any invention but drew level with their first attack in the 21st minute.
Stephen Pearson and Rob Hulse worked a smart one-two on the left side of the area and when Pearson was halted by a challenge, on-loan Arsenal teenager Sunu poked the loose ball past Adam Federici from the edge of the area.
Hulse then had the ball in the net again but was offside while Deeney made his first save for Derby as he rushed out to block Bertrand's effort.
It was a brave stop but counted for little as Kebe put the hosts back ahead in the 34th minute.
Jay Tabb played the ball through and after left-back Jay McEveley made a hash of intercepting the pass, Kebe was clear to sprint through and finish.
Kebe then hammered a volley over from a Long cross as Reading looked for a killer third before Deeney saw red in the 41st minute.
Paul Green was at fault for underhitting a back-pass and Tabb raced through before being hauled down by the debutant.
Savage took the gloves to the delight of the home crowd and watched on as Long smashed the penalty miles over the crossbar.
Savage kept the keeper's jersey for the second half and dashed out to clear from Long.
Savage then had the whole ground on their feet with a magnificent save to tip over a dipping free-kick from Gylfi Sigurdsson before he was finally beaten in the 69th minute.
Bertrand cut in from the left and hammered in a shot from 15 yards which was deflected cruelly past Savage - who had no chance.
Substitute Church made it four from close range with six minutes left after Kebe had again skinned the defence to give the Royals three valuable points.