Does Islam offer eternal salvation?

I know that Islam recognizes the Torah and Jesus, but have a question about salvation I have been pondering for a while. I was raised Christian and don't doubt my belief in God, but I don't see how Jesus ties in with Mohammed (excuse my ignorance.) The Bible says we don't need to earn salvation because it was given to us by the grace of God. This is why Jesus died for our sins. In Islam, does one need to earn salvation? Are they guaranteed a place in heaven if they live a holy life? What determines this? Thanks for any input=)