The Danish centre-half was accused of "ball-watching" as Michail Antonio stole in to put West Ham into a 2-1 lead in their Premier League clash on Wednesday.
Speaking to Sky Sports after the game, Cole said: "He's got to recover and block that space.
"He recovers too late and he doesn't see where Antonio is. Now he's ball-watching. It's a great ball and now it's a tap-in.
"Whenever you recover to your goal you've got to have an open body shape, to be able to see the ball and to see the striker maybe out the corner of my eye but he did nothing. He just looked at the ball.
"He left so much space between the goalkeeper and the centre-half. It's a great ball and for Antonio it's a tap-in.
"There's nothing Azpilicueta can do because he's got a man on him as well, he can't track that run.
"Christensen has to start realising the danger and the spaces before he just switches off for a second."