Nine-time winner Fanja had won the first-leg game 1-0 in Salalah on February 15. Al Nasr joined Saham, Al Khabourah and Salalah in the last-four stage.
Fanja had last won the coveted championship in 2014, while had emerged winner way back in 2006.
Earlier on Sunday, unfancied Al Khabourah shocked two-time winner Al Suwaiq 2-1 at the Sohar Sports Complex to make its maiden entry into the HM's Cup.
The teams were involved in a goalless draw in the first leg played on February 15. On Saturday night, Salalah edged past Al Rustaq 5-4 following a tie-breaker to advance to the last-four stage in Salalah.
Salalah had won the away leg 1-0, but Rustaq drew parity on Saturday to force the game into extra-time.
The teams ended in a 2-2 stalemate in regulation-time.