NATO chief Jens Stoltenberg said Sweden would make the alliance "stronger and safer" while the United States, the main alliance power, as well as Britain and Germany welcomed Sweden's now imminent accession.
German Chancellor Olaf Scholz said that having Sweden in NATO "strengthens our defence alliance and with it the security of Europe and the world".
Russia's invasion two years ago prompted Sweden and neighbouring Finland to apply to join the trans-Atlantic bloc, ending their longstanding stance of non-alignment.
Every NATO member has to approve a new country however, and Hungary's vote ended more than a year of delays that frustrated the other 31 nations as Ukraine battled Russian troops.
Finland joined in April last year, but Sweden's bid was stalled by both Hungary and Turkey, with Ankara approving Stockholm's candidacy only last month.
Hungary then followed, with 188 parliament members voting in favour and six far-right deputies against.
"Today is a historic day... Sweden stands ready to shoulder its responsibility for Euro-Atlantic security," Sweden's Prime Minister Ulf Kristersson said on X.
Speaking about Russia's potential reaction, Kristersson told a press conference: "The only thing we can expect with any certainty is that they don't like Sweden becoming a member of NATO, nor Finland".
Going forward, "Nordic countries will have a common defence for the first time in 500 years... we remain friends, and we become allies," he said.
- 'Mutual good intentions' -
Hungary's Prime Minister Viktor Orban had long stalled Sweden's membership but told parliament that it would "strengthen Hungary's security".
Though repeatedly saying it supported Swedish membership in principle, Hungary kept prolonging the process, asking Stockholm to stop "vilifying" the Hungarian government.
After a meeting on Friday between the nationalist Orban and Kristersson in Budapest, the Hungarian leader announced that the two had clarified "our mutual good intentions".
Hungary also signed a deal to acquire four Swedish-made fighter jets, expanding its fleet of 14 Jas-39 Gripen fighters.
Hungary's president is expected to sign the law within days. Sweden, which has been militarily neutral for two centuries, will then be invited to accede to the Washington Treaty and officially become NATO's 32nd member.
All Baltic nations except Russia will now be part of the alliance.
Prime Minister Giorgia Meloni of Italy, which currently presides the G7 group of industrialised democracies, said Sweden's entry "reinforced NATO for the defence of peace and freedom on the European continent".
Alongside its move into NATO, Sweden signed an accord in December that gives the United States access to 17 Swedish military bases.
The looming membership has been accompanied by a toughening of declarations by its leaders. General Per Micael Buden, commander-in-chief of the Swedish military, said in January that Swedish people "must mentally prepare for war".
"It is the last piece of the puzzle in the NATO map for northern Europe," said Robert Dalsjo, an analyst for the Swedish Defence Research Agency (FOI).
People in Sweden mainly cheered the approval.
Jimmy Dahllof, 35, said Sweden would be "safer... bringing us closer to our European neighbours".
"I am very relieved because we have been waiting so long," said Ingrid Lindskrog, a 73-year-old pensioner.
In Hungary's delay, some experts saw a strategy to wring concessions from the European Union, which has frozen billions of euros in funds because of the nationalist government's policies.
Others argued it underlined Orban's closeness to the presidents of Russia and Turkey.
For Mate Szalai, an analyst at Venice's Ca' Foscari University, Orban was simply playing to his domestic audience.
"Orban wanted to go as far as he could without causing serious problems to the trans-Atlantic community while proving that Hungary is a power to be reckoned with," he told AFP.
Many of his acts are intended to provoke Europe, Szalai added.
burs-ros-jza/tw/js
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