Swedish Prime Minister Ulf Kristerson and Hungarian Prime Minister Viktor Orban met in Budapest on Friday.
Sweden has lifted the final hurdle to joining NATO after Hungary's parliament voted to approve the bid.
The Nordic country applied to join the defense alliance after Russia's full-scale invasion of Ukraine in 2022.
Each member must approve a new annexation, and Hungary has delayed, accusing Sweden of being hostile.
But last week Hungarian Prime Minister Viktor Orbán said the two countries were now “ready to die for each other”.
All NATO members are expected to help an ally under attack.
Swedish Prime Minister Ulf Kristerson said it was a “historic day” and a “big step” for Sweden to abandon 200 years of neutrality.
“Sweden is a great country, but we are joining NATO to better protect everything we are and everything we believe in,” he said.
NATO Secretary General Jens Stoltenberg said the Hungarian decision made the alliance “stronger and safer”.
Parliament's approval must now be signed by the president – after which a formal invitation is sent to Sweden to join the 31-member group.
The process usually takes a few days.
Mr Orban is a nationalist politician with close ties to Russian President Vladimir Putin. He has often blocked EU attempts to send military aid to Ukraine.
Sweden is one of the EU countries that has accused Hungary of backing away from the EU's democratic principles.
In turn, Zoltan Kovacs, Mr Orban's spokesman, accused the authorities in Sweden of sitting on a “degenerate throne of moral superiority”.
Last week, however, Mr Orban hosted his Swedish counterpart Ulf Kristerson and announced his support for Sweden's membership.
The vote by Hungarian MPs on Monday was almost unanimous – 188 to 6.
In his speech, Mr Orban criticized unnamed NATO allies for putting pressure on his government to end the 21-month delay.
“Hungary is a sovereign country and will not tolerate being dictated by others on the content or timing of the decision,” he said.
Turkey was the other NATO country to block approval of Sweden's application over its so-called support for Kurdish separatists. It eventually lifted its veto in January.
Sweden and its eastern neighbor Finland, long considered militarily neutral, have expressed interest in joining NATO in May 2022.
Finland formally joined in April last year, doubling the length of the alliance's border with Russia.
Russian President Vladimir Putin has sent his army to Ukraine in 2022, which is expected to check NATO's expansion and weaken Western collectivism.
In fact, in addition to Sweden and Finland, the opposite happened.
