The ceasefire allowed the road out of the city to be cleared of mines and humanitarian aid to be delivered.
Ceasefire measures will remain in place until the end of Sunday 1 November.
Residents had a four-hour window to leave before a convoy of 25 buses, with space for 750 people, left the town on Friday afternoon - along with ambulances for people with limited mobility.
Several hundred people are understood to have decided to stay in the war-torn city, despite widespread devastation and the lack of gas, electricity and running water.
Some of the older generation did not want to leave, one relative told the BBC.
Lubinets said on Friday morning that those who decided to stay "will receive all the necessary humanitarian assistance".
He and his Russian counterpart Yana Lantratova travelled to Oleshky together on Friday, setting off from Armiansk, in the Russian-occupied Crimean peninsula.
Lubinets is the first Ukrainian official since the start of the war to visit Russian-occupied parts of Ukraine.
Zelensky praised him and his team for helping move evacuees out to safety.
The Ukrainian president also thanked those who had helped bring attention to the "true humanitarian catastrophe in Ukrainian territory temporarily occupied by Russia".
The ceasefire has required close cooperation between Lubinets and Lantratova, which Lubinets said he carried out in the interests of saving lives.
But some in Ukraine have recoiled at the sight of him standing beside a representative of the Russian occupying power which has killed so many of the city's residents and made life unbearable for the rest.
Russia said on Thursday that it had prepared "dozens of tonnes" of aid for the residents who do not wish to leave, including food, medicine, hygiene products and fuel for generators.
Additional reporting by George Wright