Alonso, a big-money summer signing from Liverpool, believes the capital club should have taken something from the game.
"I feel like we could have done better," he told Real Madrid's website. "We deserved more. We had more chances, especially in the first half. It's a shame.
"It was a neck-and-neck battle from start to finish. Perhaps they had more possession, but we had better opportunities. They didn't feel comfortable on the pitch.
"It's a pity we couldn't score first because I am sure the outcome would have been much different. We played close together and knew how to move the ball quickly in attack, but all we missed out on was a goal."
Despite seeing his side overtaken at the summit, Alonso knows there is plenty of time to make amends and is already looking ahead to Saturday's visit of Almeria to the Bernabeu.
"It was an important match, but it wasn't decisive," he said. "There is a long season ahead and each match is another step towards the goal. Now we must focus on Almeria."