Alaves ceded the initiative to the Blancos for the most part in the early going at Mendizorrotza Stadium in this northern city and focused on generating chances on the counter-attack late on Saturday.
The visitors then dominated the action and had two clear chances to open the scoring before the intermission, but a shot by Gareth Bale hit the woodwork after ricocheting off Alaves defender Ximo Navarro in the 10th minute, reports Efe news.
The first score came amid driving rain early in the second half on a perfectly executed set piece, with Toni Kroos sending a free kick into the area and Sergio Ramos timing his move perfectly and heading the ball inside the far post in the 52nd minute.
The goal completely altered the complexion of the match, as a freshly determined Alaves stepped up their aggressiveness in search of the equalizer.
The hosts clamoured for a penalty on a possible hand ball by Croatian star Luka Modric in minute 56 and then won an opportunity from the 12-yard mark seven minutes later when Ramos was called for elbowing Joselu in the face inside the area.
On the penalty try, Lucas Perez delivered the ball to one corner as Areola was diving the other way to make the score 1-1.
But the celebration didn't last long at Mendizorrotza.
In the 69th minute, Dani Carvajal scored on a tap-in shortly after a high corner kick was poorly defended by Alaves.
Real Madrid (31 points) have provisionally moved into first place with the victory and have a game in hand on all of the other teams except arch-rival FC Barcelona.
( With inputs from IANS )