Los Blancos managed to come back from a goal down at half-time but failed to secure all three points, leaving them one point behind La Liga leaders Barcelona.
We lacked a bit of accuracy with the chances we had to take all three points," said Alonso at the post match press conference.
"But we're still in it [La Liga title race], it's all very close. It's a long season and we have to keep going."
The Spanish giants have played three consecutive draws in the league since a 4-0 win against Valencia early last month.
"The team had the desire to turn the 1-0 deficit around. We had three or four clear chances inside the area to score the second goal and win," added the Spanish head coach.
"We have to continue with the high standards required to play for Real Madrid. It's a long season."
Madrid will be back in La Liga action on Wednesday when they face Athletic Club.