"Today is a good day for Europe," Swedish Prime Minister Fredrik Reinfeldt said in a statement released after partial results showed some 65 percent of voters in favour of the treaty.
"It has been a long journey. Now the presidency will be active to reach all the way."
Ireland received several guarantees, following the original No vote, that the treaty would not affect key policies such as its military neutrality or its abortion and tax laws.
"Europe has listened to - and acted on - the concerns of the Irish people. This is European cooperation at its best. It's a good thing for Ireland and a good thing for the European Union," Reinfeldt said.
The Czech Republic and Poland are the only two EU nations yet to sign off on the treaty designed to streamline the workings of the bloc which has increased from 15 to 27 nations since 2004.
Reinfeldt said that "It is now important to get the treaty in place. The European Council is united in its wish to see the Treaty enter into force before the end of the year."
Poland will sign the treaty "shortly," Reinfeldt said, adding that he would also meet Czech Prime Minister Jan Fischer and EU Commission chief Jose Manuel Barroso in Brussels on Wednesday.
"We will then discuss the situation and see what actions can be taken to move the situation forward."