Abbas, a moderate from the rival Fatah Party, immediately distributed the money to a variety of humanitarian programmes agreed upon with Russia, his office said.
The transfers were made with the knowledge of the Hamas-controlled finance ministry, officials said.
Palestinian officials said the delivery of the aid could serve as a model for a European proposal to set up a fund, administered by the West, to reach humanitarian aid to the Palestinians without dealing with Hamas.
The so-called Quartet, which includes the US, European Union, Russia and the United Nations, is expected to discuss the proposal on Tuesday.
The EU and the US, the two largest donors to the Palestinians, have cut off hundreds of millions of dollars in direct aid to the Hamas-led Palestinian Authority, because they both have branded Hamas, which refuses to recognise Israel as a legitimate state, a terrorist group.