The European Medicines Agency recommended the vaccine developed by US pharma giant Pfizer and German firm BioNTech for use, and the European Commission formally approved it hours later.
The EMA added that the vaccine would "very likely" be effective against a new strain of the disease spreading through Britain.
European Commission chief Ursula von der Leyen said vaccinations would start across the EU on December 27, adding that the vaccine was a "true European success story".
"This is a very good way to end this difficult year and finally start turning the page on Covid-19," von der Leyen said in Brussels.
German Health Minister Jens Spahn said the decision allowed a "road out of the crisis" while Italian Health Minister Roberto Speranza said the EMA decision "is the news we have been waiting for".
- 'Historic scientific achievement' -
The Amsterdam-based EMA, the drugs watchdog for the 27-nation EU, had moved the decision forward from December 29 under pressure from EU governments, particularly Berlin.
"It is a significant step forward in the fight against this pandemic that is causing suffering and hardship," EMA chief Emer Cooke told an online press conference as she announced the decision to recommend the vaccine.
"This is really a historic scientific achievement, within less than a year a vaccine will have been developed and authorised against this disease."
The urgency surrounding the virus has increased with the news that a fast-spreading variant is sweeping Britain, prompting a growing number of countries worldwide to suspend flights from the UK.
But EMA officials said they believed the Pfizer-BioNTech jab would be effective against it.
"At this moment there is no evidence to suggest this vaccine will not work against the new variant," Cooke said.
The EMA's head of vaccine strategy, Marco Cavaleri, added that while they were waiting for more data "for the time being we are not too worried".
"It is very likely that the vaccine will retain protection also against this new variant," he said.
"What would scare us is if we see multiple mutations", particularly on the "spike" that the virus uses to enter human cells, but those had not been seen yet, Cavaleri added.
- 'Cause for concern' -
The EMA said it took longer than Britain -- the former home of the agency -- and the US because they used a special, short-term emergency authorisation.
The "conditional marketing authorisation" issued on Monday however lasts for one year and required more rigorous testing, it said.
The EMA also had to contend with a cyberattack in which data from the Pfizer-BioNTech and Moderna vaccines was stolen.
Cooke said the agency had "worked night and day" to speed things up, but needed to make sure the Pfizer-BioNTech vaccine was safe and effective, in order to avoid any doubts that could affect uptake.
"We know very well that the speed at which these vaccines were developed and authorised is a cause of concern for many Europeans," said Harald Enzmann, chairman of the EMA committee that took the final decision.
But he said the authorisation followed "one of the largest trials we have ever evaluated for a vaccine" and that it "met the standards for robustness and quality that we have set out".
The authorisation is for over-16s only and says that the vaccine should be given to pregnant women on a case by case basis, the EMA said.
Following a "small number" of reports of allergic reactions in Britain and the United States, the EMA had recommended that people should be kept under "close observation" for 15 minutes after vaccination.
A European decision on another vaccine, produced by US firm Moderna, is due by January 6.
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