The Norwegian side, who are currently in the middle of their own domestic campaign, had plenty of defending to do, but the visitors just could not find a way through in Oslo.
Madrid, who were without injured duo Cristiano Ronaldo and Karim Benzema, saw one more opening come and go before the interval - Danilo breaking into the box but seeing a poor final touch let him down.
A much-changed visiting side emerged for the second half but, while they continued to ask questions of their opponents, they were struggling to find a way through.
Bale again found himself in the right place to try another weak attempt, which Burchert was able to keep out, shortly before being taken off.
A lively Denis Cheryshev then curled an effort narrowly wide as the Spanish giants closed in on a goal, while the Valerenga keeper again had to be alert to deny Borja Mayoral, but there was to be no breakthrough for Real.