So, it is quite possible, although by no means certain, that Jesus was aware of some of the key ideas and teachings of the Buddha. Indeed, several of Jesus' key teachings and sayings [eg 'Love your enemies and pray for those who persecute you' (Mt 4:44); 'The kingdom of God is within [sic] you' (Lk 17:21)] are quite non-Semitic but entirely consistent with Buddhism.
Now, it has been written, ‘No anattā doctrine, no Buddhism.’ Indeed. The concept of anattā is bedrock to Buddhism. Anattā means ‘no-self’ or, more correctly, ‘not-self.’ The Buddhist teaching of anattā affirms that there is no actual ‘self’ at the centre of our conscious - or even unconscious – awareness. Our so-called consciousness goes through continuous fluctuations from moment to moment. As such, there is nothing to constitute, let alone sustain, a separate, transcendent ‘I’ structure or entity. We ‘die’ and are ‘born’ (or ‘reborn’) from one moment to the next.
As I see it, that is how Jesus ‘saves.’ He shows us the way out of the hell and the prison we have made for and of ourselves. He shows us how we can be relieved of the bondage of self and thereby gain true freedom and happiness. Not only that, he lived out the truth of ‘not-self’ in his own life and, even more importantly, in his death on the Cross. Powerful stuff.