Whereas the Holy Spirit is instrumental in the increase of Christ, the Cross is instrumental in the decrease of Self. Remember that the whole trouble with man is his inherent tendency to travel a path that is independent of his Creator. This is why the Cross must do a very deep, thorough work in every disciple of Jesus.
As Self is submitted to Christ, Sin begins to lose its power of attraction, and Satan has no more leverage. The night He was betrayed, Jesus told His disciples, “The ruler of this world is coming, and he has nothing in Me” (John 14:30). That is to say, there was nothing in the Lord Jesus that the devil could use to get an advantage over Him. This is the condition He desires for each of His disciples, and the Cross is the means through which that condition is arrived at.