Pugh had been linked to North End a couple of times in the past year or so, after failing to cement a regular place in the Leeds side.
The 23-year-old made 57 appearances for the Whites scoring 6 goals. He started his career at Manchester United and was Kevin Blackwell's first major capture as Leeds boss.
Simpson said: "He is a player I have always liked so I wasted no time in making my interest known. I am absolutely delighted we have got him.
"He is at a great age, with plenty of football in him and he's of very good pedigree with his grounding at Manchester United.
"He has a lot of quality and a great left foot and can play in either position on the left side."
Pugh has agreed a three-year deal but as the winger was technically a free agent after reaching the end of his contract at Elland Road, United are entitled to compensation, as he is under the age of 24 and recently rejected the club's offer of a new two-year deal.