The country has never played before in a major football competition. But it is not yet clear whether Albania will qualify for the next round in France.
The team will get €1m (£770,000; $1.1m) in extra funding as well as the new passports, the government said.
Albania lost their other two matches, coming third in their Euro 2016 group.
Albanian Prime Minister Edi Rama captured the nation's excitement on Sunday by tweeting "Goooooooooool...." when Armando Sadiku headed the winning goal against Romania in the first half.
The award of diplomatic passports is mainly symbolic, as many of the Albanian players are already with clubs elsewhere in Europe and Albanians enjoy visa-free travel to the EU.
They will not enjoy diplomatic immunity, as that is reserved for official diplomats, but they will benefit from easier international travel.
The win sparked big celebrations in the streets of Albania's capital Tirana and in Pristina, capital of neighbouring Kosovo, where most of the population are ethnic Albanian.
Fans waved Albanian flags, let off fireworks, honked car horns for hours and danced to folk music.
The European football body Uefa has launched disciplinary proceedings against the Albanian and Romanian football federations because of crowd trouble at the match.
Some supporters let off smoke bombs, hurled objects and invaded the pitch. One French-Albanian man hid a banned flare in his rectum to smuggle it into the stadium, prosecutors said.
Uefa has now opened disciplinary actions against eight countries in the tournament because of hooliganism by fans.

Crowds gathered outside the presidential palace on Saturday morning and then marched to parliament.
Inside parliament itself, opposition MPs have continued a sit-in that began on Friday.
Polish Prime Minister Beata Szydlo, in a televised address, called the MPs' actions "scandalous".
She said people were free to protest, but had to respect the views of others.
Poland's populist right-wing government wants to limit the number of reporters allowed to cover parliament.
But opposition MPs accuse it of trying to stifle press freedom.
Amid a heavy police presence, a crowd of about 2,000 people gathered outside the presidential palace on Saturday chanting "freedom, equality, democracy". Some held up copies of the constitution.
The crowd later marched to the parliament building where they remained throughout the day.
European Council President Donald Tusk, a former Polish prime minister, urged the government to respect the constitution on Saturday.
Addressing reporters in the western city of Wroclaw, he criticised the government's plans, saying that without media access "democracy becomes dictatorship".
In extraordinary scenes on Friday, opposition MPs blockaded the parliamentary plenary chamber, forcing MPs from the PiS party into another room to vote on next year's budget.
It was the first time since the restoration of democracy in 1989 that such a vote was held outside the main chamber of parliament.
Outside parliament, several thousand protesters gathered overnight on Friday. Police had to forcefully remove people to allow MPs to leave the building.
Protester Szymon Roginski said on Saturday that the confrontation was entering "a new, more aggressive phase".
"Every day we hear news that makes us understand that we are further and further away from democracy. People have had enough," he said.
Leader of the opposition Nowoczesna party, Ryszard Petru, accused the government of usurping parliament's authority.
"They [the government] do not allow journalists [access], they close themselves off and meet in other places and call it the parliament," he said.
"This is an usurpation of power and there will be no consent from the opposition or Polish society for it. We will protest both at the parliamentary podium and on the streets of Polish cities."
PiS has been accused of restricting press freedom since coming to power last year.
Next year only a few reporters will be allowed into parliament, with five selected TV stations permitted to make recordings of parliamentary sessions.
PiS leader Jaroslaw Kaczynski said on Friday that the proposals were no different to the media access in many other European nations.
He accused protesters of hooliganism and threatened them with unspecified "consequences".
Albania says it will reward its national footballers with diplomatic passports for having beaten Romania 1-0 at the Euro 2016 tournament.
[NEXT_CONCEPT]
Demonstrators in the Polish capital Warsaw have renewed their protest against government plans to restrict journalists' access to parliament.