Wales defender Joe Rodon headed in Joe Rothwell's corner in the 39th minute on Saturday before Austria international Wober, who replaced injured left-back Junior Firpo in the 21st minute, tapped home in the 44th minute.
It marked the first goal either defender had ever scored for Leeds.
Those two strikes were just reward for the West Yorkshire side’s first-half dominance and provided the platform for a victory which saw them bounce back from the previous Saturday's disappointing 1-0 defeat at Blackburn.
For Paul Warne's Rams, this latest setback marked a third straight defeat and left them with just one win in their last 10 Championship games.
Leeds dominated possession during the opening quarter but initially struggled to make it pay against a hard-working Derby outfit.
In the 13th minute Leeds right-back Jayden Bogle advanced forward and tried to play in Brenden Aaronson, but his pass was blocked by a Derby defender.
The Rams continued to defend stoutly in the face of mounting pressure from their hosts.
Craig Forsyth did well to block a dangerous, low cross from the right by Dan James before Kayden Jackson was booked for hauling down Aaronson.
Another cross by James was then cleared by Curtis Nelson and in the 21st minute Leeds lost Firpo to a knock.
But the Whites continued to probe and it needed a fine block from Derby defender Eiran Cashin to deny Aaronson after he found the space to shoot on the edge of the six-yard box.
From a corner on the right moments later Rothwell's delivery was met at the far post by the unmarked Rodon, who sent a header wide from six yards out.
In the 38th minute Aaronson did well to turn inside the box and fire a shot which Jacob Widell Zetterstrom parried behind.
From the resulting corner on the right, Rothwell flighted in another fine out-swinging delivery and this time Rodon made no mistake, planting a firm header into the net from six yards out.
As half-time approached, Leeds scored their second when James' neat pass found Aaronson down the right flank.
His low, whipped cross deflected off Jerry Yates and into the path of Wober, who tapped home right-footed from six yards out to make it 2-0 at the break.
Max Wober tucks away Leeds' second goal in their comfortable win over Derby on Saturday.Max Wober tucks away Leeds' second goal in their comfortable win over Derby on Saturday. (Image: Danny Lawson/PA Wire.)
Six minutes into the second half Manor Solomon curled a right-footed shot just wide of the far post from 20 yards out.
In the 58th minute James hit a thumping 25-yard effort which was parried by Widell Zetterstrom, who then dived low to his right to keep out Ao Tanaka's 20-yard strike.
Moments later the overworked Rams goalkeeper did superbly to turn Joel Piroe's low shot around a post for a corner.
Liam Thompson then hit a 25-yard shot at Leeds goalkeeper Illan Meslier in what was Derby's first effort on target as Leeds ran out comfortable winners.