The Tirana stadium was plunged into darkness by a power cut at the start of the second half of the match and the Real players, including Kaka and Cristiano Ronaldo, had to wait for an hour and 20 minutes for the electricity to be restored.
Real had been invited to Albania to play newly-promoted first division side Gramozi, who had borrowed players from other teams to boost their chances.
Gramozi are owned by oil businessman Rezart Taci and his company paid Real more than two million euros for their 10-hour trip to Tirana, in what critics said was an unwelcome addition to a busy schedule for the Spanish team.
Angry fans shouted and threw lighters on to the pitch when the lights went out. Unconfirmed reports said the stadium manager had been sacked over the incident.
Real's Karim Benzema scored the winning goal in the 80th minute after Gramozi's Daniel Xhafa had put the home side ahead in the third and Kaka had equalised in the 37th thanks to a cross from Ronaldo.
Captain Raul was the only Real player to stay for the presentation of a cup, before leaving in a hurry surrounded by bodyguards.
"At least all went well. The match took place, thank God," said former Albania international turned television commentator Sulejman Starova.
"This is a warning that we should improve our single real stadium."
