The UEFA congress voted 28 to 24 with two votes declared invalid to accept Kosovo, which will also have a team at the Olympics for the first time at the Rio de Janeiro Games in August.
FIFA will now decide at its congress in Mexico City on May 13 whether to become the last major sporting federation to admit Kosovo.
The Kosovo Football Federation hopes that if FIFA gives its green light, it will get a late place in a European qualifying group for the World Cup in Russia in two years.
"Now we have to wait and see what FIFA decides in 10 days in Mexico, but we hope for the best, that we can compete in the World Cup."
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There were also new celebrations in Kosovo where the congress was broadcast live on four major TV channels and by radio stations.
President Hashim Thaci told AFP: "After three decades of isolation for Kosovo football, Kosovo's membership in UEFA marks a new page, a historic day.
"I strongly believe that the next step will be Kosovo's membership in FIFA," he added.
Serbia's Football Association president Tomislav Karadzic said it would open up a "Pandora's Box" with other breakaway regions applying for membership.
But Vokrri said UEFA had to vote for footballers.
"What message are we sending the youth of Kosovo," the federation president told the congress before the vote.
"Sixty percent of the population of Kosovo are aged under 27. Can we really keep them away from this privilege."
Switzerland, which has several ethnic Kosovo players in its national team, including Xerdan Shaqiri, Granit Xhaka and Valon Behrami, also opposed the vote.
European football officials did not rule out action by
opponents of the vote at the Court of Arbitration for Sport.
Kosovo was at the centre of a war between ethnic-Albanian and Serbian forces in 1998-99 and declared its independence from Serbia in 2008.
It is now recognised by more than 100 countries, but it is not a member of the United Nations.
The Olympic appearance will be another victory in its battle for international recognition while the prospect of at least taking part in European club and international tournaments could see a new force emerge.
UEFA membership is only the first stage of getting into the Champions League or the European Championship however.
Its stadiums and clubs will have to be approved by UEFA's club committee and then the UEFA executive will have to football to let Kosovo onto the international stage.
UEFA could also decide to keep Kosovo and Serbian teams apart in European competitions.