He did not say specifically where the territories were located and provided no timeline, except to say he had received “good reports” at a Sunday meeting of his military commanders and intelligence chief.
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 Russia’s energy dependence is being used against Europe by Moscow.
Last week, Moscow said it would maintain the closure of the Nord Stream 1 gas pipeline – its main gas channel to Germany – and the G7 countries announced plans to cap Russian oil export prices. The Kremlin has declared that it will stop selling oil to all countries that apply this cap.
Moscow blamed Western sanctions and technical problems on energy disruptions. European countries that have backed kyiv with diplomatic and military support have accused Russia of weaponizing energy supplies.
Also on Sunday, the US Embassy in Moscow said that Trump-appointed John Sullivan left Russia after completing his term as US ambassador to that country.
A US State Department official said Mr Sullivan had “a typical length of stay for US ambassadors to Russia”.