Both sides headed into the contest on the back of opening-game wins, but it was the visitors who emerged with all three points thanks to another impressive display.
Picking up from where they left off against Stoke City six days ago, United were soon in front thanks to Klich's curler just inside the post after Max Lowe lost possession high up the field.
Fikayo Tomori wasted a good headed chance on his first start since joining from Chelsea as Derby pressed for a quick response, and they had it seven minutes after going behind.
Bielsa's men turned on the style, however, with Roofe leaping highest to meet Alioski's cross for his first of the evening.
Leeds gave themselves some breathing space with an hour played when Roofe, moments after being denied by a fine Scott Carson save, took a touch to get himself away from Richard Keogh and blasted high into the net.
Alioski added a fourth with 25 minutes still to play, crashing a header past Carson following some good play from Pablo Hernandez.
A fifth should have followed, only for Saiz and Luke Ayling to squander good chances, although Mount also had a shot kept out by Peacock-Farrell in the final 15 minutes.
DERBY COUNTY: Carson; Lowe, Tomori, Keogh, Wisdom; Mount (Bennett 75'), Ledley (Johnson 46'), Bryson; Lawrence, Waghorn, Wilson (Jozefzoon 61')
LEEDS UNITED: Peacock-Farrell; Ayling, Berardi, Cooper, Douglas; Hernandez, Saiz (Baker 87'), Phillips, Klich (Shackleton 73'), Alioski; Roofe (Bamford 79')