The drones flew more than 1000km to hit the terminal, Zelenskiy said on social media on Wednesday.
Clouds of black smoke rose over the city's port after the attack.
Russian authorities said only that the Ukrainian drone strike targeted the city's infrastructure, without providing further details.
The airport of St Petersburg briefly suspended flights overnight because of the attack.
Authorities also cut off mobile internet services.
With the front line changing little as swarms of drones hinder battlefield movement, both sides have sought an edge by increasingly launching long-range strikes.
The war that followed Russia's invasion of its neighbour has stretched into its fifth year, with no end in sight.
The latest strikes are an embarrassment for Putin, weeks after he in May pruned back an annual Victory Day parade in Moscow due to fears of Ukrainian drone attacks.
Putin is set to speak on Friday at the economic forum in St Petersburg that the Kremlin views as a prestige event.
The gathering has for decades been Russia's leading event for attracting foreign capital.
It is sometimes called Russia's Davos, likening it to the World Economic Forum held in Davos, Switzerland.
Major Western investors and officials have stayed away since Russia invaded Ukraine more than four years ago.
Saudi Arabia is a special guest country and is due to send a large business delegation.
The strikes came a day after Russian forces launched a massive drone and missile attack on Kyiv and other Ukrainian cities, killing at least 22 civilians and wounding 138, as Moscow followed through with its threat of escalating its regular barrages.
Kremlin spokesman Dmitry Peskov said Russia's deep strikes had already taken on a "systematic" character.
Ukraine is short of American-made Patriot air defence missiles, in part because of US stocks depleted by the Iran war, leaving it vulnerable to Russia's ballistic missiles.
NATO chief Mark Rutte arrived in the Ukrainian capital Kyiv on Wednesday for talks with officials that likely will cover air defence needs.
Ukraine's own long-range attacks are aimed at diminishing Russia's oil production, which is a key source of funding for Moscow, and disrupting weapon production.
Ukraine has repeatedly targeted oil facilities at the port of St Petersburg and nearby ports.
Ukrainian drone attacks overnight also hit the Kronstadt naval base, an old base for Russia's Baltic Fleet, and a manufacturing plant involved in weapon production in Russia's Tambov region, 600km from Ukraine, Zelenskiy said.
Russia's defence ministry said that air defences downed 354 Ukrainian drones overnight.
In the Russia-controlled part of Ukraine's Donetsk region, a Ukrainian strike hit a bus that was travelling from Moscow to Crimea, killing seven and injuring 11, according to the Kremlin-appointed head of Donetsk, Denis Pushilin.
In the Smolensk region, two firefighters were killed by a Ukrainian drone attack, according to the regional governor, Vasily Anokhin.
Meanwhile, Russia fired 198 long-range drones at Ukraine, according to Ukraine's air force, with air defences neutralising 189.
Authorities in Ukraine's northern Sumy region said that in the previous 24 hours one civilian was killed and 15 more were injured, including three children, by Russian strikes.
In the southern Kherson, Russian overnight shelling and drone strikes killed an 86-year-old woman and wounded five other people, according to regional authorities.
