The Nigerian striker, who joined the Foxes from CSKA Moscow earlier this summer, scored in the 48th and 67th minutes to spark the reigning Premier League champions back into life after they had trailed 3-0 at half-time.
A first-half brace from Munir, either side of a Luis Suarez finish, had Luis Enrique's Barca in control at the break before Musa turned the game on its head with a pair of goals less than 20 minutes apart.
The Foxes opener came early in the second term when Musa picked up possession in Barcelona's half before advancing into his own attacking area and finishing coolly.
The deficit was further cut when Jeffrey Schlupp found Musa from the right with the new addition firing high into the net with another quality finish.
However, any hopes of a comeback at the Friends Arena in Stockholm was put to bed when Rafael Mujica swept home in the 84th minute.