"If we adopt it as presented today at the last ambassadorial meeting, it is a significant step forward," the minister told the public broadcaster LRT on Wednesday.
"First of all, regarding oil," he added.
The fresh package of sanctions includes a ban on Russian oil imports, something opposed by Hungary.
According to Landsbergis, a compromise should be found.
"It appears that one country can hold the whole sanctions policy hostage. This is not fair," the minister said. "Of course, part of the public rhetoric is politics: Hungary is stepping up pressure on its EU partners."
"I think that the search for compromises that is now underway will be successful – I want to believe that – and we will manage to find compromises that will meet Hungary's needs and help maintain a united sanctions policy," he added.
If a compromise is found, it will send a signal to both Russia and other oil suppliers, according to Lithuania's top diplomat.
Among other things, the sixth package proposes new sanctions on banks and on Patriarch Kirill, the head of Russia's Orthodox Church, and others.