European Council President Charles Michel announced the decision - which also included starting negotiations with Moldova - as “a clear signal of hope for their people and for our continent”.
Orban - who had insisted for weeks he would block the move - agreed not to be in the room when the other leaders made a consensus decision.
In a video posted to social media, the veteran leader denounced “a completely senseless, irrational and wrong decision” but complained that “26 other countries have insisted this decision be taken”.
If he had chosen to use his veto Orban could have derailed the summit, but he backed down one day after the European Commission unlocked 10 billion euros in frozen funds for Hungary.
“Orban made his case, made it very strongly,” said Irish premier Leo Varadkar.
“He disagrees with this decision and he’s changing his opinion in that sense, but essentially decided not to use the veto power.”
Alongside the decision on Ukraine, which has been invaded and part-occupied by Russia, the EU summit also granted candidate status to Georgia - one step behind formal negotiations.
Ukraine’s President Volodymyr Zelensky, who did not attend the knife-edge summit, hailed the decision as “a victory that motivates, inspires, and strengthens”.
And Moldova’s President Maia Sandu declared: “Moldova turns a new page today with the EU’s go-ahead for accession talks. Moldova is ready to rise to the challenge.”
The White House - which faces opposition from Republicans in the United States to its own support for Ukraine - hailed a “historic decision”.
The decision to open membership negotiations with Kyiv provides a major boost for Ukraine as fears swirl over Western support, but the road remains long to actually joining the EU.
Before the talks can be launched, EU states must agree on a negotiating framework - giving Orban ample opportunity to stall the process again.
50-billion-euro question
Attention at the summit now focuses on a plan to grant Ukraine a four-year 50-billion-euro funding package from the EU budget that Orban has also threatened to block.
Kyiv desperately needs the money to prop up its war-torn economy, and to change the narrative that backing from its allies is waning.
Most EU leaders wanted this week’s summit to send a sign of solidarity with Ukraine 22 months after Russia launched an all-out invasion.
But any decisions must be unanimous - or at least unopposed - and Orban initially insisted a decision on funding could wait until after June’s European elections.
Critics have accused the Hungarian leader of holding Kyiv’s survival hostage in a bid to force Brussels to release billions of euros of EU funds frozen over a rule of law dispute.
In what some saw as a last-minute concession, the European Commission, the EU’s executive, agreed on Wednesday to unblock 10bn euros of that cash.
Another 21bn euros still remain out of Orban’s grasp, but he denied that Hungary was making a link between the cash and its Ukraine stance. “That’s not our style,” he said.
No ‘victory’ for Putin
Zelensky, in an impassioned plea via video link, earlier told the leaders “now is not the time for half-measures or hesitation”.
He said failure to open membership talks with Ukraine would be used by Putin “against you personally, and against all of Europe.”
“Don’t give him this first - and only - victory of the year,” he urged.
Beyond Orban, other EU leaders stressed the need for unity and to send a strong signal of support for Ukraine, which has already seen Washington’s support threatened by manoeuvres in the US Congress.
The leaders said the bloc had agreed a 12th round of sanctions on Moscow, targeting Russia’s lucrative diamond exports and aiming to tighten an oil price cap.
But the situation on the battlefield in Ukraine does not look promising for Kyiv after a counter-offensive failed. Putin boasted yesterday that he has 617,000 troops in Ukraine, and that their positions are improving.
Across Brussels, at NATO HQ, alliance secretary general Jens Stoltenberg warned that the West must continue supporting Ukraine in order to protect the rest of Europe.
“If Putin wins in Ukraine, there is real risk that his aggression will not end there. Our support is not charity - it is an investment in our security,” he said.