FIFA and UEFA had previously blocked Kosovo's attempts to play internationals because it was not recognised by the United Nations.
However, FIFA President Sepp Blatter urged his executive committee to approve the change at its meeting on Monday and told Reuters when the announcement was made on Tuesday that he hoped this accelerated Kosovo's drive to full membership of the organisation.
Other sporting federations could follow suit, giving Kosovo another step towards wider international recognition.
"All the other countries that were in the Balkan organisation - Slovenia, Croatia, Bosnia, Serbia, Montenegro and Macedonia - all became independent and play within FIFA and UEFA, and the last one left is Kosovo," Blatter said.
"Kosovo has organised football, so I think it is a logical gesture and solidarity and this decision by the FIFA executive committee will accelerate the movement towards membership."
Fadil Vokri, the head of the Kosovo Football Federation, told Reuters: "We have been waiting for this day for many years and the right to organise friendly games will give a new push for our federation and our football.
"This is the last step before we will be a full member."
Not everyone was happy with the decision and the Serbian Football Association (FSS) lodged a sharp protest with FIFA.
"We strongly protest the decision by FIFA's executive committee to allow the world soccer governing body's members to play friendly matches with the so-called national team of the so-called Republic of Kosovo," the FSS said on its website.
"The decision made behind closed door in Budapest without any consultations with either UEFA or the Serbian Football Association blatantly contravenes the FIFA statutes.
"Hence we will ask for an urgent meeting with FIFA President Sepp Blatter and his UEFA counterpart Michel Platini to demand a reverse of this unjustified decision which could have far-reaching consequences across the region."
The ruling means that the 90 members of the UN who recognise Kosovo as an independent state can now play them and Vokri told Beta news agency by phone: "That means we are still ineligible to play in competitive qualifiers for major events but at least we can play friendlies with countries which have recognised Kosovo."
In 1999, NATO bombed Serbia for 11 weeks to halt the killing and expulsion of ethnic Albanian civilians in Kosovo by Serb forces fighting a two-year counter-insurgency war.
After nine years as a ward of the United Nations, Kosovo declared independence from Serbia in 2008.
Among the 90 countries that recognised Kosovo were the United States and 22 of the 27 members of the European Union, Blatter said.
Serbia, backed by Russia, says it will never recognise Kosovo sovereignty, and the country has been unable to clinch membership of the United Nations, blocking its participation in many international bodies and sporting events.