"If people think that nothing depends on them or have no trust in this, there will be no future," he said in an interview with LRT TV aired on Tuesday.
Lithuania's top diplomat added that the continued existence of the EU "will require effort, motivation and leadership".
"I believe in the future of Europe. I really do. But I want to add that this bright future cannot be taken for granted," he said.
Linkevicius also called for efforts to ensure that citizens of Lithuania should all feel the benefits of EU membership.
"Statistical tables or World Bank ratings (...), we can be proud of that, but people must feet it. Achieving that takes more time and effort," he said.
President Dalia Grybauskaite's foreign policy advisor, Nerijus Aleksiejunas, told Ziniu Radijas that Lithuania had become a reliable and respected partner in the 15 years since it joined the EU.
"This has been earned by hard and consistent work," he said.
Aleksiejunas said Lithuania had made a big leap in terms of mentality and world-view, which it could not have done without European support.
When asked to comment on the president's vision of the EU's future, the advisor said it was "a safe, strong, prosperous, united, modern and solidarity-based Europe".
Lithuania joined the EU on May 1, 2004.