The two teams will only have three days to recover before they face each other again when Madrid host Barcelona in La Liga on March 2.
The first legs will be played on either February 6 or 7 - with Real hosting Barca first - with the following leg on either the 27th or 28th of the month.
"There's nothing like it", said Real Madrid director Emilio Butragueno. "It's going to be a great tie". We are talking about two teams that have a lot of resources and extraordinary players.
Arthur has played 15 league games (838 minutes) in LaLiga this season, and not one of the important fixtures has been lost.
The LaLiga giants now occupy third position in the league standings, 10 points behind table-toppers Barcelona. Four days after that, Madrid travels to the Netherlands to face Ajax in the last 16 of the Champions League.
The final will take place on May 25 at Betis' home ground, the Estadio Benito Villamarín. Betis meanwhile beat Espanyol 4-2 on aggregate, with Valencia overcoming Getafe 3-2 on aggregate by scoring twice in stoppage time in their 3-1 victory in the second leg at Mestalla.
The last time the two rivals met, Barca inflicted a humiliating 5-1 defeat on Real which led to the exit of manager Julen Lopetegui.
Real Madrid have won the competition 19 times.