And Cazorla twice laid on fine balls for his team-mates that left Madrid relieved not to concede an equaliser.
Karim Benzema equalised with a header and Raphael Varane soon gave Santi Solari's side the lead but Cazorla would have the final say as his 82th minute finish ensured a draw.
Los Blancos are now sat in fourth place in the La Liga table, but they can move level on points with third place Sevilla with a win on Thursday.
"We had some chances to kill off the game", Courtois said.
Real are now fourth in the LaLiga table, eight points behind Barcelona, taking on the Yellow Submarine on Thursday before hosting Real Sociedad on Sunday.
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Real's Casemiro became the first player to be booked in the match four minutes before time with a late tackle.
"We started off badly with an error and a goal, but then we got back into it".
"Every player here has to be committed, dedicated and keep working with hope and excitement; they are all fundamental to us", Solari told a news conference ahead of Real's rescheduled game away to Villarreal on Thursday.
"We needed those three points to close the gaps on the leader".
Coach Luis Garcia, who replaced Javier Calleja before Christmas, is without suspended right back Mario Gaspar and is likely to play the defensive Javi Fuego in central midfield.
Villarreal receive Madrid in the relegation zone, but know that a draw will be enough to lift them above Athletic Club on goal difference.