Real Madrid moved three points clear at the top of La Liga with a hard-fought 2-1 win over Alaves on Saturday.
Sergio Ramos had given Zinedine Zidane's side a lead minutes after the restart at a rain-soaked Mendizorrotza Stadium by heading home a Toni Kroos free kick.
The Real Madrid captain then gave away a penalty from which Lucas Perez levelled in the 65th for the Spanish striker's eighth goal in nine games.
Real took the lead four minutes later as Isco pounced to head a Luka Modric cross against the post and the ball trickled across to Carvajal, who bundled it in from close range.
Second-placed Barcelona will look to move back level on 31 points with with Real Madrid on Sunday when they travel to fourth-placed Atletico Madrid.
"We know that in stadiums like this teams always give everything against Real Madrid, and we coped very well with the kicks, the rain and the cold which was crucial because these are the games which win you league titles," said Real Madrid left-back Marcelo.
A 1-0 defeat at Mendizorrotza 13 months ago came three weeks before Julen Lopetegui was sacked as coach but this time Madrid showed the kind of steel that was so painfully lacking last season.
Alaves, who had lost only one of their previous seven home games this season and conceded only two goals on their own turf before facing Real, are 13th on 18 points.
Real were without Eden Hazard after he injured his ankle in Tuesday's thrilling 2-2 draw with Paris Saint Germain in the Champions League, leaving Gareth Bale to make his first start for the club since October 5.
Zidane made further changes by leaving World Cup-winning defender Raphael Varane at home and fielding Brazilian Eder Militao in his place.
Militao very nearly gave away a penalty in the opening minutes when he dangled out a leg on Aleix Vidal in the area.
The referee took a dim view of Vidal's theatrics though and booked the Spaniard for diving instead.
Bale, who has been the subject of whistles and jeers from Madrid fans in recent weeks after holding up a playful flag during international duty for Wales, came close to giving the visitors the lead when he saw a header deflected on to the near post, while Isco sent a shot narrowly over the bar as the first half drew to a close.
It was far from a free-flowing display from Zidane's side but the type of win that will give them confidence that they can overhaul an inconsistent Barca side in what is set to be a close-run title race.
