No, my answer is maybe.
First of all, I'm not a universalist. There will be some that go to hell and there will be some that go to heaven. And you are absolutely right, there is a narrow road.
I believe in the born again experience. I believe that Jesus is the Way, the Truth, and the Life, no man can come to the Father, but by Him. But I also believe that not everyone who professes the name of Christ is truly born again, and the Bible backs this up:
"Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.
Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works?
And then will I profess unto them, I never knew you: depart from me, ye that work iniquity." - Matthew 7:21-23
So what is the will of the Father? Well, skip over some chapters in Matthew:
"Then shall the King say unto them on his right hand, Come, ye blessed of my Father, inherit the kingdom prepared for you from the foundation of the world:
For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in:
Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me.
Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed thee? or thirsty, and gave thee drink?
When saw we thee a stranger, and took thee in? or naked, and clothed thee?
Or when saw we thee sick, or in prison, and came unto thee?
And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me." - Matthew 25:34-40
This is the kind of person God is looking for. And you might ask, how is this different that the works that are described in Matthew 7? It has to do with the attitude of the heart. God is not looking for people who do good works, but people who will love, people who will sacrifice their lives in compassion and mercy for people.
When someone finds this kind of love in their heart, they have found God, they have found Jesus, they have found His Holy Spirit. For the Way, the Truth, the Life that Jesus said He was encompassed this kind of love and compassion. The whole point of the Law was to love God and love thy neighbor.
I contend that there are those in different religions that have found life in Christ, though they might not realize it. I've heard story after story of those whose lives were changed to make them compassionate people. Those who find this love are born again. It is the fruit of the Spirit.
What of Christ's sacrifice? The purpose of sacrifice was to atone for sins and bring people back into a relationship with God. And while I believe that Christ died for our sins, I'm open to the possibility that that application could apply to those who are repentent of their sins regardless of who and what they are, if they seek God with all their heart and seek to get things right. It is a universal sacrifice.
Having said this, I still believe that it is better to follow Jesus in the light that you have. That as one gains the knowledge of Christ through scripture, that person can grow. But I also believe that those lacking knowledge, but are seeking God, will be taught by the Holy Spirit how to live and how to love.
You only need to go to the example of the Good Samaritan to see the truth in this. This Samaritan was not a Jew, in fact the Jews despised and ostrecized the Samaritans because they didn't believe as they do. They have erred in doctrine. They weren't considered true Jews, yet in this story, this Good Samaritan is the one who had compassion on his neighbor, even for his enemy.
Remember, Jesus gave this parable in response to a certain lawyer's question of how to inherit eternal life:
"And, behold, a certain lawyer stood up, and tempted him, saying, Master, what shall I do to inherit eternal life?
He said unto him, What is written in the law? how readest thou?
And he answering said, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy strength, and with all thy mind; and thy neighbour as thyself.
And he said unto him, Thou hast answered right: this do, and thou shalt live." - Luke 10:25-28