The 56-year-old Scot earned the position by a majority vote of the EPO's executive committee members. Dallas will oversee the selection and assessment of referees and their appointments for matches.
The move comes after continued pleas from Greek clubs for a foreigner to take on the administrative role following several match-fixing scandals in recent seasons.
Dallas retired from refereeing in 2005 having officiated at two World Cups (1998 and 2002), the 1996 Olympic Games, the 1999 UEFA Cup final and several Champions League matches.
From June 2009 to November 2010 he was head of referee development at the Scottish FA.