The Scot went into game under severe scrutiny after seeing his side take just seven points from their previous 11 games in the SkyBet Championship.
But his side looked anything but one short of confidence against the in-form Rams and after a nervous start tore them apart with an impressive display of attacking football.
Oliveira started the ball rolling on 14 minutes and went to complete a perfect three timer – left foot, right foot, header – with a quick double late in the second half.
Derby could have no complaints but they did not help themselves when Jacob Butterfield got himself sent off midway through the second half when the score was 1-0.
Norwich were on the back foot in the opening stages, with Derby winning an early free-kick in a dangerous position on the edge of the box which Matej Vydra curled wastefully wide.
There was an edgy atmosphere in the Carrow Road stands after City’s recent poor run and the hosts were not helping themselves, with a string of misplaced passes increasing the tension.
But that all changed on 14 minutes when the Canaries scored with their first effort on goal. The Rams were guilty of backing off as Oliveira picked up possession midway inside the opposition half and the Portuguese striker needed no second invitation to take aim and blast home a low shot into the bottom corner from some 20 yards.
Norwich nearly doubled their lead on the 20-minute mark after a clever interchange on the right between Oliveira and Pinto ended with Steven Naismith volleying the latter’s cross just over.
Scott Carson then needed to make a smart block to deny Jacob Murphy after the young winger had latched onto a superb through ball from the recalled Wes Hoolahan before denying Naismith following an assist from the same source.
With the confidence flowing back Martin Olsson was the next to test Carson with a shot from a tight angle while Alex Tettey fired a low shot against an upright as the hosts pushed for a second.
Derby had done well to get to the break just one goal down and they were soon under pressure after it, with Oliveira clipping the crossbar after claiming a loose ball on the right.
It was still all Norwich and Oliveira wasted a glorious opportunity to make it 2-0 after 56 minutes when he blasted high and wide after Murphy’s cutback had given him a clear sight of goal.
Derby were still very much in it at 1-0 however and almost made Norwich pay for their wastefulness when Tom Ince wriggled his way through before being denied by an excellent save from the underworked John Ruddy.
But there chances of getting anything from the game suffered a severe setback midway through the half when ex-Canary Butterfield saw red for a reckless lunge at Hoolahan near the halfway line.
Murphy then failed to take another presentable chance to make the points when he screwed the ball wide from inside the area while Jonny Howson saw an effort palmed away by Carson.
The Canaries finally got the second goal they deserved on 77 minutes. A period of sustained pressure ended with Russell Martin’s cross from the left being nodded on by Howson for Oliveira to provide the finishing touch.
And less than two minutes later the striker completed his hat-trick when he thumped home a loose ball from just inside the box after a ragged Derby had failed to clear their lines.