Aivazoff's four-year career with the Victoria Cougars in the WHL established him as a prospect of note, and the L.A. Kings selected him 109th overall in the 1988 draft just before his final year in which he reached the 100-point mark.
Life got tougher when he attended an NHL training camp, though, and he spent the next four years in the minors, signing with the Red Wings as a free agent in 1993. After a year with Detroit he was claimed by Pittsburgh in the Waiver Draft and then claimed again by Edmonton just a few minutes later when the Penguins exposed him in the same draft.
Aivazoff saw little action with the Oilers the next year, signed as a free agent by the Islanders, and spent most of 1995-96 in the IHL. Although he was again signed, by the Rangers, he never returned to the NHL, playing the next four of five years in the "I" with the exception of 1997-98 when he played half a year in Germany.
In 2000-01 Aivazoff returned to Germany, suiting up for the Schwenningen Wild Wings of the DEL where he would play two seasons before retiring following the 2001-02 season.