The Swiss finance ministry said that Britain had received a payment of 372 million francs ($331 million, 310 million euros), and Austria, 515 million.
The money was paid on behalf of accounts which the owners did not want to reveal to their own tax authorities.
The ministry also revealed that 15,000 British clients and 13,500 Austrians – with combined holdings of 12.8 billion francs – had opted to come clean to their own countries' authorities.
A ministry spokesman said the payments were just the first in a series of monthly instalments running until June 2014.
With the financial crisis having put Switzerland under mounting pressure to lift its trademark banking secrecy, the country has opted to give ground in some areas in order to defend the overall principle of discretion.
Under bilateral deals with Britain and Austria, Switzerland has offered two options to people who fail to declare in their home countries money placed in Swiss banks.
They can either turn themselves in to their homeland's revenue services, or have their accounts taxed by the Swiss, who then transfer the funds without naming the clients.
It was under the latter system that Switzerland handed over the sums revealed on Thursday.
The ministry said that the results of the deal showed that it could achieve its goal of having a financial sector that met tax requirements, while the Swiss bankers' association said the figures for those who turned themselves in were in line with forecasts.
Similar deals are being negotiated with Italy and Greece.
Such accords are specifically related to past wrongdoing, and are separate from another deal under which Switzerland taxes European citizens' accounts and transfers the money to the individual's homeland.