The names of those released have not been confirmed by the Government, while few details are yet known about how their release was secured, along with five other people.
The Foreign Office had been working for months to support those detained, while it is believed that Crown Prince of Saudi Arabia Mohammed bin Salman was involved in helping to free the British nationals.
Ms Truss, who is visiting New York for a UN summit where world leaders are discussing the ongoing war in Ukraine, tweeted: "Hugely welcome news that five British nationals held by Russian-backed proxies in eastern Ukraine are being safely returned, ending months of uncertainty and suffering for them and their families."
She thanked Ukrainian president Volodymyr Zelensky "for his efforts to secure the release of detainees, and Saudi Arabia for their assistance".
"Russia must end the ruthless exploitation of prisoners of war and civilian detainees for political ends," Ms Truss added.
Mr Aslin was one of those released, according to his local MP Robert Jenrick.
A video later emerged of two men sitting inside an airliner, in which Mr Aslin introduced himself and Mr Pinner, adding: "We just want to let everyone know that we're now out of the danger zone and we're on our way home to our families."
Mr Pinner interjected: "By the skin of our teeth", as Mr Aslin continued: "We just want everyone to know the good news etc, so thanks to everyone that's been supporting us and whatnot, so it's really muchly appreciated."
Mr Pinner added: "Thanks to everybody."
A court in the self-proclaimed Donetsk People's Republic had sentenced Mr Aslin to death alongside fellow British detainee Mr Pinner in July.
The BBC reported that Mr Pinner, from Bedfordshire, was also among the group of five set to come home, and had been photographed with Mr Aslin, while it said the Saudi Press Agency had released pictures which also showed a third British man, John Harding.
Mr Harding, along with Britons Andrew Hill and Dylan Healy, went on trial last month in the city of Donetsk, Russian media reported.
The three, along with Swede Matthias Gustafsson and Croat Vjekoslav Prebeg, all pleaded not guilty to charges of mercenarism and "undergoing training to seize power by force", according to Russian media.
The next court hearing in their case was scheduled for October, the Interfax news agency reported, citing a statement by the separatists' court.
Associated Press said the 10 prisoners included citizens of Sweden, Croatia and Morocco, as well as two US military veterans, Alex Drueke, 40, and Andy Huynh, 27.
Mr Jenrick said he was "deeply grateful" to the Ukrainian government, as well as the Saudi Crown Prince and the Foreign Office, for securing the release.
He added: "Aiden's return brings to an end months of agonising uncertainty for Aiden's loving family in Newark who suffered every day of Aiden's sham trial but never lost hope. As they are united as a family once more, they can finally be at peace."
Foreign Secretary James Cleverly said the release "brings to an end many months of uncertainty and suffering, including the threat of the death penalty, for them and their families, at the hands of Russia".
"Tragically that was not the case for one of those detained and our thoughts remain with the family of Paul Urey."
The British aid volunteer died earlier this year while being detained by Russian-backed separatists in Ukraine.
"I would like to express my gratitude to President Zelensky and his team for their efforts to secure their release, and to HRH Crown Prince of Saudi Arabia, Mohammed bin Salman and his team, for their assistance," the Foreign Secretary said in a statement.
"I continue to call on Russia to comply with International Humanitarian Law and not exploit prisoners of war and civilian detainees for political purposes."
Allan Hogarth, from Amnesty International UK, called it a "huge relief after a "sham judicial process apparently designed to exert diplomatic pressure on the UK".
"The Donetsk authorities and their counterparts in Russia are still believed to be unlawfully holding thousands of other detainees amid horrifying reports of torture, and we call on them to immediately release all such detainees."