It's less 'save' and 'not destroy' although that's maybe a fine distinction.
In Genesis God is drawn to do something about Sodom as the din of shrieks from that city has spurred him to action, shrieks either from people of the city oppressed in it or visitors to the city oppressed by it. Considering the Lot story later and God's intention to destroy Sodom presumably shrieks by visitors. Anyhow. Yes, Abraham haggles. Just prior to this we hear God's internal monologue where he decides, for the first time, to include a human -- post Adam, depending on how you read the naming of things -- in the decision making process that governs the world. Abraham is presumably horrified -- the narrator of Genesis does not say -- and haggles God down from destroying the city outright, to 50 etc etc daring to go as low as 10, a number that, just so happens, to be a later minimum administrative unit size in Jewish society. In the narrative structure of Genesis we have already seen an attempt to eradicate evil through destruction -- the Flood -- and that does not work, to the point where God promises not to do outright, global destruction like that again. So it's clear in the narrative -- though perhaps not to Abraham who may or may not have known about the post-Flood promise -- that God has a maximum upper bound on the amount of people that can be destroyed in response to evil: all. Abraham brings this maximum upper bound down to 50 as an opening gambit, then etc etc. It is worth noting that Abraham, at this point in the story, is elderly and rich, so he's presumably used to negotiation as a way of life.
Why would Sodom not be destroyed if there were less honest people? God's intention before consulting Abraham is to destroy the whole city but it is Abraham that bargains the number down to a minimum. If ten can be found, the threshold for destruction isn't reached. An entirely reasonable read here is that Abraham couldn't bear to see an entire city's worth of people destroyed and God was willing to be convinced otherwise. In itself that's a remarkable thing for a Near Eastern deity and is one sign that Genesis as a piece of literature is in conversation with and opposed to other contemporary Near Eastern literature.
Anyway, that's not the only read here -- there are millennia of commentary on this very text -- but it is worth pointing out that the Genesis text comes from the Mesopotamian culture, one that is both distinct from our Greek-derived way of thinking/being and has also gone extinct outside of literature, so norms that may have appeared self-evident to the original audience might not come through to modern readers so easily, or at all.