Paul actually answers this quite well, in Romans 7:
21 So I find this law at work: Although I want to do good, evil is right there with me. 22 For in my inner being I delight in God’s law; 23 but I see another law at work in me, waging war against the law of my mind and making me a prisoner of the law of sin at work within me. 24 What a wretched man I am! Who will rescue me from this body that is subject to death? 25 Thanks be to God, who delivers me through Jesus Christ our Lord!
So then, I myself in my mind am a slave to God’s law, but in my sinful nature[d] a slave to the law of sin.
It is only through the regenerating work of the Holy Spirit who kills the sinful nature, that we are saved. That is why being saved works - As Paul says in Galatians 2:20:
I have been crucified with Christ and I no longer live, but Christ lives in me.
That is why we must die with Christ. As Paul said earlier in Romans 7:
1 Do you not know, brothers and sisters—for I am speaking to those who know the law—that the law has authority over someone only as long as that person lives? 2 For example, by law a married woman is bound to her husband as long as he is alive, but if her husband dies, she is released from the law that binds her to him.
Once we are made alive again in Christ, however:
Romans 8:9 You, however, are not in the realm of the flesh but are in the realm of the Spirit, if indeed the Spirit of God lives in you. And if anyone does not have the Spirit of Christ, they do not belong to Christ. 10 But if Christ is in you, then even though your body is subject to death because of sin, the Spirit gives life because of righteousness. 11 And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies because of his Spirit who lives in you.
12 Therefore, brothers and sisters, we have an obligation—but it is not to the flesh, to live according to it. 13 For if you live according to the flesh, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live.
C.S. Lewis' The Great Divorce captures this really well in Chapter 11. In this chapter, we meet an otherwise agreeable ghost (one of the creatures who is visiting heaven, and being made fit to live there) who has a lizard on his shoulder. The ghost wants to be saved, but the lizard is a representation of all his sin and vice, whispering constantly in his ear, and leading him astray.
An Angel and the ghost have a conversation, in which the Angel simply and repeatedly asks, "Do you want me to kill it." The chapter counters all of the protests the ghost is making to killing the lizard, but in the end, the ghost finally allows the Angel to kill that sinful nature.
What is amazing about the metaphor is what happens when the Angel is finally allowed to kill it.
"Have I your permission?" said the Angel to the Ghost.
"I know it will kill me."
"It won't. But supposing it did?"
"You're right. It would be better to be dead than to live with this creature."
"Then I may?"
"Damn and blast you! Go on can't you? Get it over. Do what you like," bellowed the Ghost: but ended, whimpering, "God help me. God help me."
Next moment the Ghost gave a scream of agony such as I never heard on Earth. The Burning One closed his crimson grip on the reptile: twisted it, while it bit and writhed, and then flung it, broken backed, on the turf.
" Ow! That's done for me," gasped the Ghost, reeling backwards.
For a moment I could make out nothing distinctly. Then I saw, between me and the nearest bush, unmistakably solid but growing every moment solider, the upper arm and the shoulder of a man. Then, brighter still and stronger, the legs and hands. The neck and golden head materialised while I watched, and if my attention had not wavered I should have seen the actual completing of a man-an immense man, naked, not much smaller than the Angel.
What distracted me was the fact that at the same moment something seemed to be happening to the Lizard. At first I thought the operation had failed. So far from dying, the creature was still struggling and even growing bigger as it struggled. And as it grew it changed.
Its hinder parts grew rounder. The tail, still flickering, became a tail of hair that flickered between huge and glossy buttocks. Suddenly I started back, rubbing my eyes. What stood before me was the greatest stallion I have ever seen, silvery white but with mane and tail of gold. It was smooth and shining, rippled with swells of flesh and muscle, whinneying and stamping with its hoofs. At each stamp the land shook and the trees dindled
The new-made man turned and clapped the new horse's neck. It nosed his bright body. Horse and master breathed each into the other's nostrils. The man turned from it, flung himself at the feet of the Burning One, and embraced them. When he rose I thought his face shone with tears, but it may have been only the liquid love and brightness (one cannot distinguish them in that country) which flowed from him. I had not long to think about it. In joyous haste the young man leaped upon the horse's back. Turning in his seat he waved a farewell, then nudged the stallion with his heels. They were off before I well knew what was happening.
"Do ye understand all this, my Son?" said the Teacher.
"I don't know about all, Sir," said I. "Am I right in thinking the Lizard really turned into the Horse?"
"Aye. But it was killed first. Ye'll not forget that part of the story?"
"I'll try not to, Sir. But does it mean that everything-everything-that is in us can go on to the Mountains?"
"Nothing, not even the best and noblest, can go on as it now is. Nothing, not even what is lowest and most bestial, will not be raised again if it submits to death. It is sown a natural body, it is raised a spiritual body. Flesh and blood cannot come to the Mountains. Not because they are too rank, but because they are too weak. What is a Lizard compared with a stallion? Lust is a poor, weak, whimpering whispering thing compared with that richness and energy of desire which will arise when lust has been killed."
"But am I to tell them at home that this man's sensuality proved less of an obstacle than that poor woman's love for her son? [referring to an earlier story in which a obsessed mother was demanding to have her son, even though he was already in heaven] For that was, at any rate, an excess of love."
" Ye'll tell them no such thing," he replied sternly. "Excess of love, did ye say? There was no excess, there was defect. She loved her son too little, not too much. If she had loved him more there'd be no difficulty.