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Yes, it is true. We are saved by God’s grace. Whether or not one explicitly knows that is not the issue. It is by the sacrifice of the Son that we are saved. God didn’t have to do that, but He did. You might thumb your bible for the verse about “doing the WILL of the Father.” God does not just expect you to sit around on your butt. If an atheist consistently does the WILL OF THE FATHER then he too can be saved. We must pray for those who manage, by natural law, to develop an ethos of helping people, and avoiding evil. The atheist, has a harder time to be sure, but don’t assume all are going to hell.

John 14:6

Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.

Seems clear enough.
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Bingo! So right. Well, What did we learn from this papal announcement? We learned the Pope can be dead wrong.

That blows papal infallibility right out the window, doesn’t it?

Not as much as it may seem, at least not in contemporary Catholic thinking. An atheist who does good and lives a good life adheres to natural law, which is ultimately divine in origin. In that sense, one who rejects God but does good and lives a good life has nevertheless followed God's direction as to how we should live.

The Gospel also informs us that we may fully know God as He intends only by believing in and following Christ. Natural law and reason alone are not enough to attain such an understanding. Christian faith and practice are required.

This line of reasoning offers a resolution of the long and vexatious controversy as to whether those who lived before Christ or who do not know otherwise know and follow Him may be saved. Thus, although rejecting Christ forfeits the deeper knowledge of God and the grace that Christianity offers, one who adheres to the principles that God has laid out through natural law may still be saved.

As stated by Pope John Paul II, "the followers of other religions can receive God’s grace and be saved by Christ apart from the ordinary means which he has established," meaning the Christian faith. (Letter to the Bishop Delegates to the Fifth Plenary Assembly of the federation of Asian Bishops' Conferences, 23 June 1990.)

Notably, the same apostolic letter also states that "It is a contradiction of the Gospel and of the Church’s very nature to assert, as some do, that the Church is only one way of salvation among many, and that her mission towards the followers of other religions should be nothing more than to help them be better followers of those religions." The passage from John that you cite is taken as authoritative toward this end.