This is the concluding part of Philippe Sands’s extraordinary trilogy – part history, part moral investigation, part memoir – that documents the legal and personal battles to bring to account Nazi war criminals and their disciples.

In East West Street he recounted the plight of Lviv, the city now in Ukraine, whose Jewish population either fled before Nazi occupation or, like many of Sands’s extended family, was thereafter wiped out. Two Jewish lawyers who got out early were instrumental in creating the legal concepts of crimes against humanity and genocide that were introduced at the Nuremberg trials.

His follow-up The Ratline was focused on the Nazi Brigadefuhrer Otto von Wachter who died in mysterious circumstances in Rome after the war while trying to escape to South America. It is a haunting story of the feats of denial that enable otherwise “normal” people to perform or excuse the very worst deeds.

38 Londres Street follows the Nazis metastasis to its sordid death throes in Chile, the fascist safe haven Wachter failed to reach. Here we meet Walther Rauff, a former SS commander who played a key role in the development of mobile gas chambers and was directly responsible for the deaths of more than 90,000 Jews.

Having escaped from an Italian internment camp, he settles first in Syria, where he helps reorganise the country’s intelligence service, before being recruited by an Israeli intelligence officer as a Syrian informant – the Israeli was unaware of Rauff’s past and, presumably, his unabashed and lifelong antisemitism.

After a brief return to Rome, he heads for Ecuador, where he meets a Chilean officer named Augusto Pinochet, before taking his new friend’s advice and relocating to Chile. The book intertwines Rauff’s shadowy existence with Pinochet’s rise to dictatorial power, and examines the rumour that Rauff participated in the torture and disappearances that characterised Pinochet’s rule.

An epistemological problem bedevils any effort to summarise Nazi war crimes, and even those of their Chilean imitators, insofar as their scale and sadism defies understanding. But one of Sands’s strengths as a writer is that he resists the impulse to demonise. He can find signs of humanity in the most unpromising cases, which serves, paradoxically, not to mitigate but to aggravate their crimes. These perpetrators of monstrous acts did not always take the form of monsters: both Pinochet and Rauff are described in avuncular terms by some who met them.

The one time Sands’s rational composure deserts him is when he stays overnight at Colonia Dignidad, a German colony set up by a depraved paedophile called Paul Schäfer, a crony of Pinochet who enthusiastically hosted and partook in the torture and murder of the regime’s opponents. Sands barricades himself in his room with chairs and is relieved to leave the “grotesque” site of so much horror.

Running through the book is the unprecedented case, eight years after the end of his presidency, of Pinochet’s arrest in London in 1998 on a warrant issued by a Spanish judge. While Pinochet remained under house arrest for almost 18 months in the comfortable surroundings of the Wentworth estate, Sands advised the prosecution in his role as a human rights lawyer.

The legal principles and political manoeuvring that led to the granting of the old tyrant’s extradition to stand trial in Spain, a series of appeals, and finally his repatriation to Chile on dubious health grounds are not without interest but are told in rather too much detail. Sands the lawyer may have felt a professional obligation to do justice to the complexity of the wrangling, but the effect is to slow the momentum of the narrative.

By contrast the book comes vividly to life in Chile, where Sands methodically follows the various strands of Pinochet’s and Rauff’s separate histories to expose their cynical claims of innocence – Rauff stuck to the familiar line that he was only a soldier following orders, while Pinochet affected ignorance of the atrocities committed by his subordinates in what he insisted was an existential fight against communism.

The title of the book refers to a building in downtown Santiago that once housed the Socialist party but was turned into an interrogation and torture centre by Pinochet’s secret police, the DINA. There is hearsay that Rauff had some link to Londres Street, but Sands encounters shut doors, closed mouths and missing documents before eventually establishing the disturbing facts.

As with the previous two books, a personal motivation helps propel the author’s inquiry. In this instance Rauff almost certainly extinguished the life of one of Sands’s (then young) relations in eastern Europe, and there is also the plight of Carmelo Soria, a Spanish UN employee brutally tortured and murdered in 1976 who was distantly related to Sands’s wife.

These connections are not used for emotional purposes but as a means of establishing the victims as real people, with lives and family, who were killed in their thousands because of their leftwing sympathies. Part of a South America-wide and US-backed campaign, known as Operation Condor, this reign of terror aimed to crush all opposition to the continent’s rightwing dictatorships.

There were those in the Tory party, such as Norman Lamont and Margaret Thatcher, who visited Pinochet in Wentworth, who preferred to see him as a misunderstood cold warrior, despite ample evidence that he was personally involved in the selection of the victims and approved the brutal methods employed to extract information. Moreover, for all his posturing as a patriot, he stashed away in foreign bank accounts many millions of dollars that he had stolen from Chile.

But what of Rauff? He lived for much of his time at the country’s southern extreme in Patagonia, working as the manager of a crab-canning factory. He won protection from extradition in a landmark case, put together by a Jewish lawyer, and yet managed to slip back to West Germany for a visit in the early 1960s.

Myths enshrouded this odd character in his isolated cottage with his resolutely bad Spanish. Both Bruce Chatwin and Roberto Bolaño referred to him in their work, though neither was able to create a picture quite as damning as Sands achieves with his understated doggedness.

He writes of the impunity with which the old Nazi and Pinochet acted but also the immunity they managed to attain from laws they otherwise held in contempt. They may have avoided justice, but thanks to the courageous testaments of survivors and perpetrators alike, not to mention Sands’s own commendable efforts, they could not escape the truth.

• 38 Londres Street: On Impunity, Pinochet in England and a Nazi in Patagonia by Philippe Sands is published by W&N (£25). To support the Guardian and Observer order your copy from guardianbookshop.com. Delivery charges may apply