The New Church, based on the Old and New Testaments, with insights from the writings of Emanuel Swedenborg, teaches how Jesus saves, consistent with our knowledge of a loving God.
God desires for us all to go to heaven. He doesn't judge us, or condemn evildoers to hell. Those who choose evil condemn themselves to hell by choosing to withdraw from the Lord's love and mercy. The Lord God Jesus Christ saved us by showing us how to live our lives. His entire life on earth was about overcoming evils and temptations, and his death was the conclusion of that struggle. Just as we are faced with evils and temptations in our lives, Jesus struggled against those same temptations as a human. In overcoming them, He taught us the way to live.
We are saved when we open ourselves to the Lord and his love, and draw nearer to the Lord; and we do that by living our lives loving him. What does that mean? It means obeying his commandments (avoiding evil), being of use and loving others. We may have been taught that believing in the Lord is enough to save. But belief (or faith) without actively living that faith is not truly believing. Certainly, we cannot earn our way to heaven by our works. But we only truly believe or have faith when we actively work to do God's will. Love (or charity) must be united with faith in order for either to be real.
Jesus saved us, not through his death, but through his life. He overcame evil and restored a sense of balance in the world, leaving us in freedom to choose good or evil. He taught us how we should live our lives; in fact he showed us how to do it. It is only by so doing that we can be truly happy. When we die, we continue to make these choices, which determines whether we live in heaven loving God and doing his will, or turn away from him to hell. Jesus set the stage so that we are free to choose. We must do our part to choose good, which will draw us closer to the Divine. This is how Jesus saves.
This website contains a wealth of information about the New Church, and a practical, spiritual path to happiness.