Under the bespoke scheme, those foreign drivers will be able to work in the UK from now until the end of March.
Additionally, some 4,700 visas intended for foreign food haulage drivers will be extended by two months, lasting from late October to the end of February.
But the government said temporary visas were not a long-term solution and urged firms to invest in a UK workforce.
Ministers have also extended the length of temporary visas being issued to 5,500 foreign poultry workers, amid fears of a shortage of Christmas turkeys on supermarket shelves.
Previously, the government said these temporary visas would last until Christmas Eve, but the visas have been extended by a week, and will now be valid until 31 December.
Labour leader Sir Keir Starmer has called on Prime Minister Boris Johnson to recall Parliament from party conference recess, saying "emergency action" was needed.
But Mr Johnson said the UK supply chain remained "very resilient".
The temporary visa scheme for 5,000 foreign lorry drivers was originally announced a week ago when the ongoing driver shortage first began disrupting fuel deliveries to petrol stations around the UK.
On Friday, the Petrol Retailers Association said fuel supply remained a "big problem" in south-east England - and "if anything it had got worse".
From Monday, 200 military servicemen and women, 100 of them drivers, will provide "temporary" support to ease pressure on forecourts, where queues are becoming commonplace and customers frustrated.
Some of those 200 will be seen on the roads this weekend. Having completed specialised training over the past three days, many will be accompanying regular tanker drivers on their deliveries.
Health Secretary Sajid Javid said there was "enough fuel in the country, there always has been" but it had been a challenge to provide the number of drivers required.
Trade association Logistics UK estimates that the UK is in need of about 90,000 HGV drivers - with existing shortages made worse by a number of factors, including the pandemic, Brexit, an ageing workforce, and low wages and poor working conditions.
The foreign drivers eligible for visas will not be limited to the EU, but the expectation is most of the drivers will be from Europe.
Lorry drivers have said some of the conditions they face in the job were putting off younger recruits - the average age of a HGV driver in the UK is 55.
But the PM has accused the haulage industry - as well as campaign groups representing the food sector - of being too reliant on low-paid migrant workers.
In addition to offering temporary visas, the government last week announced a number of other measures aimed at limiting disruption in the run-up to Christmas and beyond.
These include increasing HGV (heavy goods vehicle) testing capacity, sending nearly one million letters to drivers who hold an HGV licence to encourage them back into the industry, and offering training courses for HGV drivers.
A survey from earlier this year suggests a number of reasons for the driver shortage