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.
Albania says it will reward its national footballers with diplomatic passports for having beaten Romania 1-0 at the Euro 2016 tournament.