Athletic, who beat champions Barcelona earlier this season, gave an astute performance in defence and also gave their hosts a few scares, with Kenan Kodro having a goal ruled out for offside and Inaki Williams forcing Thibaut Courtois to make a diving save.
Athletic coach Gaizka Garitano did not shy away from the fact that fortune had favoured his side, who have not won at the Bernabeu since 2005. "To get anything from here you have to have a bit of luck, but we ended the game very well," he said.
"I feel bad for the players because of the effort they put in. We're annoyed that we couldn't score with all the chances we had," said Real coach Zinedine Zidane.
"The ball isn't going in for us at the moment, but there's no need to be pessimistic. We've to remember this has been a good season for us. We've had a little bit of a sour ending to the year but we can't wait for 2020 to start."
The result followed a familiar pattern for Real who had chances like in the El Clasico against barcelona, but failed convert them.
Real's top scorer Karim Benzema had a shot cleared off the line in a lively first half while Toni Kroos came within inches of scoring when he waltzed through the defence and hammered a shot which Athletic keeper Unai Simon deflected on to the bar.
Earlier, powered by a glittering show from Lionel Messi, who scored 50th goal in 2019, champions Barcelona beat Alaves 4-1.
Messi goes into the winter break as La Liga's top marksman with 13 goals ahead of Real's Benzema on 12.