Zinedine Zidane's side went ahead just after the half an hour mark through a Marco Asensio header, but the hosts were level within five minutes through Pedro Tana.
Karim Benzema put Real back ahead four minutes after coming on as a substitute but Las Palmas had the last word as another substitute in 24 year-old Araujo bundled home from a delightful Marka Livaja header into his path.
Zidane stated afterwards: "What I least liked about our performance were the two goals that we conceded, but we put in a good display right until the end. We had a tough first half on Wednesday and that wasn't the case today. We played well, created a lot of chances and we just lacked that concentration in the dying stages.
"The game was under control. We won't be throwing the towel in, we'll continue to work hard. I'm pleased with the side's display. From the first minute to the last we dictated the game and we had a great number of chances. I don't think we lacked any desire."