He did not say precisely where the territories were and provided no timeline, except to say he had received "good reports" at a meeting Sunday from his military commanders and head of intelligence.
Ukraine began a counter-offensive last week targeting the south, which was seized by the Russians early in the conflict.
In his nightly video address, Zelenskiy also warned Europe that maintaining gas ties with Russia could lead to disaster, saying energy dependence on Russia is used against Europe in Moscow.
Moscow last week said it would keep the Nord Stream 1 pipeline - its main gas channel to Germany - closed, and G7 countries announced a planned price cap on Russian oil exports. The Kremlin said it would stop selling oil to any countries that implemented the cap.
Moscow blamed Western sanctions and technical issues for the energy disruptions. European countries who have backed Kyiv with diplomatic and military support have accused Russia of weaponizing energy supplies.
Also on Sunday, the U.S. embassy in Moscow said Trump appointee John Sullivan has left Russia after finishing his tenure as U.S. Ambassador there.
A U.S. State Department official said Sullivan had served "a typical tour length for U.S. Ambassadors to Russia."
