During a speech in Copenhagen, Mr Zelenskyy thanked Denmark for donating the jets and warned that other parts of Europe will be at risk of Russian military aggression if Moscow's invasion of Ukraine is successful.
“All of Russia’s neighbours are under threat if Ukraine does not prevail,” he said.
Mr Zelenskyy said on Telegram that Ukraine would receive 42 jets in total.
Denmark pledged 19 F-16s, which could be delivered around the end of the year, when pilot training lasting four to six months is completed.
However, it could take much longer to get Ukrainian squadrons battle-ready.
US Air Force Gen James Hecker, commander of US air forces in Europe and Africa, said last week that he did not expect the F-16s to be a game-changer for Ukraine.
Getting F-16 squadrons ready for battle could take “four or five years,” he said.
Danish Defence Minister Jakob Ellemann-Jensen said on Monday that Ukraine would only be permitted to use the F-16s donated by Denmark and the Netherlands on its own territory.
On Sunday, Denmark and the Netherlands announced they would supply F-16s to Ukraine, with the first six due to be delivered around New Year.
“We donate weapons under the condition that they are used to drive the enemy out of the territory of Ukraine. And no further than that,” Ellemann-Jensen said.
“Those are the conditions, whether it's tanks, fighter planes or anything else,” he said.
Russia's ambassador in Denmark issued a statement condemning the announcement. "The fact that Denmark has now decided to donate 19 F-16 aircraft to Ukraine leads to an escalation of the conflict," Russian ambassador Vladimir Barbin said. "By hiding behind a premise that Ukraine itself must determine the conditions for peace, Denmark seeks with its actions and words to leave Ukraine with no other choice but to continue the military confrontation with Russia."
The Netherlands has 42 F-16s available, but has yet to decide whether they will all be donated.
Mr Zelenskyy said on Sunday the planes would strengthen Ukraine's air defences and help its counter-offensive against Russian forces.
Washington announced its approval of the F-16 transfers on Friday, and training of Ukraine pilots is set to begin this month, which may allow Kyiv to start using the jets in early 2024.
Three months ago the Netherlands and other countries, including the UK, said they would build an international coalition to provide fighter jet support for Ukraine.