The change will come into force at 04:00 BST on Monday.
Currently, only people who received their jabs in the UK can avoid quarantine when arriving from amber list countries, except France.
The UK government said the rule change would help to reunite family and friends whose loved ones live abroad.
Transport Secretary Grant Shapps said it would apply to people who have been fully vaccinated with a jab approved by the EU or US, with the final dose at least 14 whole days before arrival.
Travellers will still need to take either a lateral flow or PCR test pre-departure and a PCR test on the second day after they arrive.
Under-18s will be exempt from isolation, and some will not have to test, depending on their age.
It come as a further 27,734 cases were reported in the UK, bringing to an end a seven-day run of falling case numbers. Wednesday's case figure was, however, down significantly from a week ago, when 44,104 cases were reported.
Another 91 deaths within 28 days of a positive test were also reported.
Tougher rules will continue to be in place for France, which is on the amber list but still requires travellers to quarantine when they return, even if they are fully vaccinated.
Mr Shapps said this advice would be reviewed at "the end of next week" as part of the rolling assessment of travel rules.
As part of the changes, international cruise ships will be able to depart from England from 2 August - after a 16-month pause.
* People fully-jabbed in the EU or US will not need to isolate when coming to England, Scotland and Wales from an amber list country
* The change is introduced from 04:00 BST on Monday
* Travellers will still take a pre-departure test, and a PCR test on or before day two after arrival
* They will need to have taken a vaccine approved by the European Medicines Agency, or US Food and Drug Administration
* International cruises can depart from England once more, with new guidance
Scottish Transport Secretary Michael Matheson said the change to the rules would provide "a boost for the tourism sector and wider economy while ensuring public health is protected".
He added that the change would be "carefully monitored by clinicians and kept under close review".
The Welsh government said the move posed "clear public health risks" - but its shared open border with England made it "ineffective" to have different arrangements.
Welsh health minister Eluned Morgan said vaccines would reduce the risks but only if they were effective against any new variants of concern and cautioned against international travel for non-essential reasons this summer.
In Northern Ireland, people vaccinated in the UK and travelling from amber list countries have not needed to self-isolate since 19 July, subject to testing.
The Stormont Executive is set to consider the policy of allowing those vaccinated in the US and EU to enter Northern Ireland without quarantine on Thursday.
* Children under 18 who are ordinarily resident in the US and EU will be exempt from isolation
* Those under five years old do not need to test at all
* Children between the ages of five and 10 will only need to complete a PCR test on day two after arrival
* And 11 to 17-year-olds will need a pre-departure test and a PCR test before or on day two
Airlines UK, which represents big carriers, said the move would offer "a lifeline for thousands of businesses reliant on international inbound travel".
Joss Croft, of trade association UKinbound, said it would "allow the £28bn inbound tourism sector, which supports over 500,000 jobs across the UK, to finally restart".
Although US citizens will soon be able to avoid quarantine in England, Scotland and Wales, they are being urged not to travel to the UK by their country's health protection agency, the Centers for Disease Control and Prevention.
And the US border is currently closed to the UK, as well as many other countries, except for US citizens.
The UK and US have set up a taskforce to discuss a travel corridor, although earlier this week the White House said it had no plans to lift Covid-19 travel restrictions for non-Americans.
But Prime Minister Boris Johnson told LBC on Wednesday that "we're talking to them the whole time".
For 70-year-old Patricia Duncan in Spain, the changes will mean she can travel to England and be on hand to support her daughter ahead of an operation - without needing to isolate.
"I have another friend with mental health issues who is desperate for me to visit," she says.
"Normally I travel three or four times a year to see my family and friends, and it's been a long time. So I'm very glad that is happening."
But Fiona Clarke in Portugal says the cost of Covid testing is still a barrier for her to visit family in Brighton.
"I think we'll look very carefully, we'll wait and see what happens with the testing," she says.
Labour deputy leader Angela Rayner said she had "real concerns" over the plan - saying: "Each individual US state does things differently," she said. "They don't have a National Health Service that has a vaccine programme like we do with the certifications.
"So we're really concerned about making sure that new variants do not come into the UK and that we do have a system that identifies where we have variants of Covid where infection is and we're able to isolate it."