Full Disclosure: I am very pro-Japan and I used to live in Tokyo. It is full of humble, hardworking, and polite people and is a magnificent country. That being said, WW2 was a time full of harsh realities that are almost unfathomable in modern times, but they were what they were:

It saved lives because given the circumstances at the time, any alternative action would have resulted in more Japanese deaths.

If you think the two 1945 nuclear bombs cost a lot of lives, you would be shocked to find out what a joint land invasion from Soviet and US troops would do. Even if the US did NOTHING, said "lol let's go home you guys are allllright!" and just quit the war, USSR continues their invasion of Japan. The USSR had already refused to enter negotiations for conditional surrender. Also, the soviet soldiers are not reputed for showing discretion during WW2 when invading their enemies. The amount of rapes and murders in Germany are estimated to be so high that even the most generous estimates are incredible in scope.

The only way you can argue that the bombs did not end the war is if you accept the narrative that Japan was going to surrender. This is pure fantasy. Not only did Japan refuse to surrender after the 1st bomb, but former PM General Tojo advised the Emperor to persist in the war even after the 2nd. When the Emperor made his final radio address (first one, as well) to tell the people to surrender, there was an armed military resistence to take the recording to prevent it from being played so the war could continue. Japan showed no signs of imminent surrender and all those involved, both US and Japan, including the emperors own words in his radio address, say that the nuclear bombings were THE direct cause for the surrender of Japan. He could have been lying for no reason, but this is the case. Soviets made a separate war declaration against Japan and seeing the writing on the wall that they could not defend themselves having their steel and munitions manufacturing crippled by the strategic nuclear strikes, the Emperor made the choice to surrender to the US and pray for a merciful transition of power.

IF the war persists, USSR will continue their invasion. If war persists, the mass starvation of Japanese children continues even if both sides were to refuse to invade for some reason. And, anything other than unconditional surrender means war within 10 years like what we saw in Korea.

So, that's how they saved lives. It was total war. Every action meant deaths and you are responsible for the lives of thousands of people for every little thing you do. Shooting a man in peace time is evil, but shooting a man in war time to save the lives of 20 other men is the kind of decisions that WW2 leaders were faced with every single day the war persists.

And on a side note, the "Ready to Surrender" argument is worthless, because so what if Japan was ready to surrender... they still had not surrendered. They were warned that the nuclear test was successful and they would be using the weapons on mainland Japan to cripple the war infrastructure. But, they didn't surrender... so the war goes on. And, if you are Truman and the lives of millions of young American men hang in the balance, how do you look a mother who lost her young teen son in the eyes and say "we could have ended the war months ago virtually in an instant, but Japan was totes ready to surrender!" It was the only responsible action... end the war ASAP or face a land invasion and an eventual proxy war with a new antagonistic Socialist Japanese state.

Obviously, the power of nukes today is so strong and what we know about nuclear radiation, it is not acceptable to use them anymore. They exist as a deterant. And a powerful one, at that. So far, few inventions have saved more lives than nuclear bombs. They prevented WW3... but clearly they are so powerful that they have the destructive potential to be our undoing as a race. They should never be used... but at that time? Their use saved the lives of millions by taking away Japans ability to even pretend to be able to fight, even if their will remained iron.

TL:DR: Sometimes there are harsh truths. The fact nuclear bombs saved lives is a testament to how horrifically fucked up the whole war was, but the facts are facts.