The narrow gate = Jesus.
Where do we find the narrow gate in scripture:
“Enter by the narrow gate. For the gate is wide and the way is easy that leads to destruction, and those who enter by it are many. For the gate is narrow and the way is hard that leads to life, and those who find it are few.
He went on his way through towns and villages, teaching and journeying toward Jerusalem. And someone said to him, “Lord, will those who are saved be few?” And he said to them, “Strive to enter through the narrow door. For many, I tell you, will seek to enter and will not be able. When once the master of the house has risen and shut the door, and you begin to stand outside and to knock at the door, saying, ‘Lord, open to us,’ then he will answer you, ‘I do not know where you come from.’ Then you will begin to say, ‘We ate and drank in your presence, and you taught in our streets.’ But he will say, ‘I tell you, I do not know where you come from. Depart from me, all you workers of evil!’ In that place there will be weeping and gnashing of teeth, when you see Abraham and Isaac and Jacob and all the prophets in the kingdom of God but you yourselves cast out. And people will come from east and west, and from north and south, and recline at table in the kingdom of God. And behold, some are last who will be first, and some are first who will be last.
“Truly, truly, I say to you, he who does not enter the sheepfold by the door but climbs in by another way, that man is a thief and a robber. But he who enters by the door is the shepherd of the sheep. To him the gatekeeper opens. The sheep hear his voice, and he calls his own sheep by name and leads them out. When he has brought out all his own, he goes before them, and the sheep follow him, for they know his voice. A stranger they will not follow, but they will flee from him, for they do not know the voice of strangers.” This figure of speech Jesus used with them, but they did not understand what he was saying to them.
So Jesus again said to them, “Truly, truly, I say to you, I am the door of the sheep. All who came before me are thieves and robbers, but the sheep did not listen to them. I am the door. If anyone enters by me, he will be saved and will go in and out and find pasture. The thief comes only to steal and kill and destroy. I came that they may have life and have it abundantly."
The one and only way to Salvation is through Christ Jesus. He lets in those who recognize His voice, those He knows (i.e. the faithful).
He says nothing in any of those passages about the gate being opened for on us based on our works.