The findings echoed a statement on Monday by the Organisation for Security and Co-operation in Europe, whose experts identified “a significant number of serious human rights violations” during a visit to Ukraine in March.
However, Russia criticised the UN report, which it said lacked any semblance of objectivity, and accused its authors of following “political orders”.
Ms Pillay said: “Those with influence on the armed groups responsible for much of the violence in eastern Ukraine [must] do their utmost to rein in these men who seem bent on tearing the country apart.”
The 34-strong monitoring mission’s report, which covers the period from 2 April to 6 May, said police and local authorities in eastern Ukraine had connived in illegal acts and the takeover of towns by armed groups, undermining the rule of law and human rights guarantees.
Ukraine is preparing to hold presidential elections a week tomorrow, and the monitors said fair and democratic elections would be an important factor in helping to calm the situation.
However, several candidates had reported intimidation and attacks, and the monitoring mission said it had concerns about their security.
The report said there had been a “wave of abductions and unlawful detentions of journalists, activists, local politicians, representatives of international organisations and members of the military”.
It said the UN monitors had been trying to verify reports of abuses by Ukrainian government forces, and that it had credible reports of people being detained by the army in a way that amounted to “disappearances”.
In Crimea, it expressed concern about the treatment of journalists, sexual, religious and ethnic minorities, Aids patients and people who had not applied for Russian citizenship.
Crimean citizens who have not obtained the right to residency by January could face deportation from the peninsula, which Russia has annexed, it said.
The Russian foreign ministry said the report had demonised pro-Moscow separatists while ignoring “the crudest violations of human rights by the self-proclaimed Kiev authorities”.
A statement from the ministry said the report’s authors had tailored information from sources “to fit preconceived conclusions: the justification of the Kiev junta and the demonisation of the protest movement in the south-east”.
Russia has accused the United States and the European Union of helping Ukraine’s current leaders topple former president Viktor Yanukovich, and denies western allegations that it supports the separatists in the east.
Moscow does not recognise the legitimacy of the new Kiev leadership and refers to it as a junta.
Gianni Magazzeni, head of the UN human rights office’s Americas, Europe and Central Asia branch, said there was no evidence to justify concern for Russian-speaking people in Ukraine.