In Noah's case, they were saved FROM water; the flood, which was sent, as punishment, to cleanse the earth of wickedness.
Like I said, if baptism saved, there would have been no need for Jesus to have even come, never mind die in agony. John was baptising for the forgiveness of sins before Jesus began his ministry.
I would imagine it is possible to go through any Christian ritual without actually being born again and believing what you are doing. We have had many infant baptisms at our church where the parents and christening party turn up, make some vows and you never see them again. Granted, that could be for any reason, and the Lord can still minister to/challenge them. But if baptism saved, then their child would, at the point of their baptism, be saved and reconciled to God. So there would be no need for repentance, accepting Jesus and being born again later on. The child could grow up to be an atheist, cult member, criminal etc etc - and yet they were saved at baptism.
It was Jesus' death on the cross that reconciled us to God, Romans 5:10-11; 2 Corinthians 5:18-20.
Jesus was the Lamb of God who takes away the sins of the world, John 1:29, and the Good Shepherd who laid down his life for the sheep, John 10:11.
There is no other way to God, John 14:6, and no one else who can save us, Acts 4:12.
We have every spiritual blessing IN CHRIST, Ephesians 1:3.
A person can be saved, born again, filled with the Spirit and enjoy all these blessings before they are baptised - and if they died before baptism, they would still be saved and forgiven.
Jesus saves us and washes away our sins.
Baptism is important - it shows us, and others, what Jesus has already done for us. When I was baptised as an adult, I first gave testimony to how Jesus had saved me and what he had been, and was, doing in my life. A school friend, who wasn't a Christian at all at school, met Christ at university, was saved and born again. She later publicly declared her faith and was then baptised.
Neither she, nor I, entered the water as an unsaved sinner and rose as a full believer; I don't know of anyone who did.
Like I said, if baptism saved, forgave and gave us every blessing in Christ, people would be saved as babies and have no need to confess Christ as adults.
Baptism comes from a word meaning "to emerge". What happens in adult baptism is symbolic of what has happened to us spiritually. For Jews, water represented death; especially the river Jordan. (This idea is reflected in literature such as Pilgrim's Progress and hymns, e.g "when I tread the verge of Jordan".) Being submerged in the Jordan was symbolic of dying. Dying to what? Sin. When someone was/is raised up out of the water again, it is symbolic of rising to new life. When we accept Christ and trust that his atoning death has reconciled us to God, we become born again; die to our sinful nature, become new creations, 2 Corinthians 5:17, put on Christ and are filled with his Spirit.
All this can, and often does, happen before a person is baptised.
Usually also, in my experience, new believers want to be baptised into a church fellowship, the local body of Christ, by their minister, or maybe by the person who led them to Christ. All of this takes time to arrange. They may have been a Christian for a few months before this can happen; saved at the point of receiving Christ, not at the time of baptism.