The home side produced a trademark defensive performance after snatching an early lead through midfielder Saul Niguez.
The result gives Madrid a slender advantage to take with them to Munich next week.
Later tonight, the semi-finals begin in the Europa League.
Liverpool have made the trip to face Villarreal in the first-leg of their semi-final clash.
And Shaktar Donetsk will host another Spanish side, Sevilla in the Ukraine.
Both matches will kick-off at five-past eight.
