Straw welcomed the start of a process that may take a decade or more, saying Turkey's entry would "bring a strong, secular state that happens to have a Muslim majority into the EU - proof that we can live, work and prosper together".
Ankara has worked for more than four decades to join, restructuring its legal system and economy to meet European standards even as Europe added demands and refused to start formal negotiations.
The agreement late on Monday night was a hard-won victory for the government of Prime Minister Recep Tayyip Erdogan of Turkey, who has staked his political credibility on getting them under way.
He hailed the beginning of talks, saying: "Turkey has taken a giant step forward on its historic march."
But the bitter struggle over the terms of the talks reflects Europe's deep ambivalence towards Turkey's membership.
Many Europeans - more than half according to some polls - oppose Turkey's membership, arguing that while the country has a toehold in Europe, it is not European.
Austria did not say why it backed
down from its membership policy
Critics say the union would have difficulty absorbing such a large, poor country and complain that Turkey's membership would open the doors for a potentially huge wave of Muslim immigrants.
Agreement talks were delayed after Austria demanded they should include an alternative to full membership, giving the union a diplomatically palatable option to inviting Turkey to join.
Vienna eventually gave up on its demands, but it may have been appeased by the likelihood of Croatia's European membership talks restarting within days.
Frozen since March over the country's poor cooperation in arresting a fugitive war crimes suspect, Austria has championed its neighbour's EU membership cause.
On Monday, the chief prosecutor of the UN war crimes tribunal, Carla Del Ponte, told European foreign ministers that Croatia was now cooperating fully - a sharp reversal of her assessment a few days earlier during a visit to the Croatian capital, Zagreb.
Turkish Nato concern
But after an initial breakthrough, the agreement was then blocked by Turkey's objections due to fears it could force Ankara to support an effort by the Republic of Cyprus to join the North Atlantic Treaty Organisation (Nato).
Turkey withdrew its objections after Secretary of State Condoleezza Rice assured Erdogan that his negotiations with Europe would not affect Turkey's voting power in Nato.
"Once Turkey enters in the European Union, all these circles will also see themselves, one way or another, represented within the EU"
Turkish foreign minister
Supporters of Turkey's membership in the US say bringing Turkey into the European club would help spread democracy into the Middle East and increase regional security.
The sentiment was echoed by Turkey's Foreign Minister Abdullah Gul before he boarded a plane in Ankara on Monday night to fly to Luxembourg.
"Once Turkey enters in the European Union, all these circles will also see themselves, one way or another, represented within the EU," Gul said.