The leader said he believed the assassination attempt was triggered by the country’s ongoing reform.
Mr Zelensky added in a statement it is “necessary” to protect the country from powerful businessmen who have “corrupted” its political system for decades.
However, his opponents say they are fearful the measure will be applied selectively to concentrate more power in the president's hands.
The law, which passed a first reading in July, provides a definition for an oligarch and bars those who fall under it from funding political parties or taking part in the privatization of state assets.
It describes oligarchs as individuals who own significant financial assets and media outlets, and requires them to register as such.
Top officials, including the president, prime minister and head of the central bank, would also be expected to declare any dealings they had with them.
Thursday's second reading of the incoming legislation, which passed with 279 votes in the 450-seat parliament, means it now goes to Mr Zelensky for approval.
The president’s team suggested any resentment towards it could be behind the attempted murder of Serhiy Shefir, a top aide and close friend of the president.
Mr Shefir's car came under attack on Wednesday, when unidentified individuals fired bullets at the vehicle as he travelled between two villages outside Kiev, Ukraine’s capital. He was unharmed though his driver was wounded, officials confirmed.
Police are searching for the weapon and interviewing possible witnesses who were picking mushrooms nearby, Ukraine’s interior ministry spokesperson Artem Shevchenko said on Thursday.
In a Facebook post, Mr Shevchenko called it a real assassination attempt and “not staged”.
Under the new law, the national security and defence council, headed by President Zelensky, has the power to designate someone as an oligarch.
Opposition lawmaker Oleksiy Goncharenko, from former president Petro Poroshenko's party, said the law “creates huge scope for corruption”, and compared it to tactics used by Russian president Vladimir Putin to increase his powers.
Kira Rudyk, the leader of the Voice party, said the bill was designed “only to strengthen the power, strengthen the position of the president and make it so that he can, together with the national security and defence council, actually decide who can have control over the media and who cannot”.
Not everyone is as cynical, though.
“[The president] is trying to distance himself from all of the oligarchs and change the vicious system in which six or seven people controlled the politics and the business in all of Ukraine,” Volodymyr Fesenko, a Kyiv-based political analyst, and head of the Penta Center think tank, told the AP news agency.
Mr Zelensky won a landslide election in 2019 during which he promised to tackle corruption and curb the influence of oligarchs, who have dominated the business landscape since the fall of the Soviet Union in 1991.
Additional reporting by agencies
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