The 24-year-old finalised his £1.3m move after passing a medical at Thorp Arch yesterday and he is in line to make his debut in Saturday’s Championship opener against Burnley.
Leeds agreed terms with Brentford and Dallas on Friday but his transfer was held up while Dallas travelled to Northern Ireland to attend a family funeral over the weekend.
It was put on hold again yesterday after power shortages at Griffin Park delayed completion of the paperwork but he became United’s seventh signing of the summer this morning.
His three-year contract includes the option of a fourth season at Elland Road.
Dallas - a Northern Ireland international - is linking up with United head coach Uwe Rosler for the second time having joined Brentford from Irish club Crusaders while Rosler was manager at Griffin Park in 2012.
He made 75 appearances for Brentford, including 28 in the Championship last season, and scored 10 goals.