Narbekovas made his debut in 1983 with Žalgiris Vilnius, Lithuania's sole representative in the Soviet Top League, at the age of 18. In 1987, he finished second in league scoring with 16 goals while leading his club to a third-place finish, their best in history. Zalgiris would then participate in the UEFA Cup for the first time, losing to Austria Vienna.Austria would become Narbekovas' destination after Lithuanian players were allowed to move abroad. Arminas moved there in 1990 after a short stint with Lokomotiv Moscow, since Lithuania was not a part of UEFA yet and players from its clubs were not permitted to transfer. Narbekovas would spend the rest of his club career in Austria, with Austria Vienna until 1995 and then with Admira Wacker and a number of other lower division clubs.
Although he never received a FIFA-sanctioned cap for the USSR national team, he did play for them, and win the Gold Medal, at the 1988 Summer Olympics. Narbekovas scored two goals in the tournament, including an extra time one in the semifinals against Italy, helping the Soviets to a 3-2 victory. Also, he represented USSR in 1987 Summer Universiade. Arminas first played for Lithuania in its first ever game as a newly independent country, on 27 May 1990 against Georgia, and scored the first goal in the 2-2 tie. Unfortunately, injuries limited his career to just 13 caps and four goals for his country.
Narbekovas was named Lithuania's footballer of the year four times, from 1985 to 1988. In November 2003, to celebrate UEFA's jubilee, he was selected by the Lithuanian Football Federation as the country's Golden Player - the greatest player of the last 50 years.