Haji-Sheikh attended the University of Michigan from 1979 to 1982 as a placekicker for the Wolverines, and in 1986 earned a Bachelor of science in geology. While attending Michigan, he set a Big Ten record by successfully converting 76 consecutive extra points. He also set the school records for career extra points (117) and field goals (31).
Haji-Sheikh was selected by the New York Giants in the ninth round (237th overall pick) of the 1983 NFL Draft. He spent three seasons playing for the Giants. As a rookie in 1983 he was successful in 35-of-42 field goal attempts (83%). His 35 field goals in 1983 broke Jim Turner's NFL record for field goals in a season. Haji-Sheikh's record stood until 1996. A recurring hamstring injury hampered the rest of his career.
In 1986, Haji-Sheikh joined the Atlanta Falcons after three seasons with the Giants. He appeared in six games for the Falcons, was successful on 9-of-12 field goal attempts, and converted 7-of-8 extra points.
Haji-Sheikh was released by the Falcons in late August 1987. In mid-September 1987, he signed with the Washington Redskins after an injury to the Redskins' regular placekicker Jess Atkinson. He appeared in 11 games for the Redskins during the 1987 season, was successful on 13-of-19 field goal attempts, and converted 29-of-32 extra points. He won his only career Super Bowl that season, kicking six extra points and missing one field goal in Super Bowl XXII. He appeared in the 1984 Pro Bowl kicking one field goal and adding six extra points. He finished his career with 76-of-111 field goals (68%) and 95-of-103 extra points, scoring 323 total points.
In 1984, Haji-Sheikh married Detroit native and fellow University of Michigan graduate Michele Blondin. As of 2016, the couple have five children. He now works in the luxury car business in Birmingham, Michigan, coaches travel soccer with Magic Soccer and trains upcoming kickers via his business Haji-Sheikh Kicking.