Mark Drakeford said the country has reached a threshold where 20 people in 100,000 are suffering from coronavirus.
But if the rate falls and Covid-19 is “effectively suppressed”, the Welsh Government will revisit the issue, he said.
Wales will make indoor meetings of more than six people illegal from Monday, though children under the age of 12 will not count towards that total.
Mr Drakeford told BBC Breakfast: “We said that, if coronavirus reached a certain threshold in Wales, we would revisit our advice on face coverings.
“Today, for the first time, we will go to a point where 20 people in 100,000 in Wales are suffering from coronavirus.
“That’s the threshold we use for people to have to quarantine coming back into the UK and, having reached that threshold, today we will be making the use of face coverings mandatory in shops and enclosed public spaces in Wales.
“But if the figure falls in the future and the virus is effectively suppressed below that 20 threshold, then once again we will be able to revisit that advice.”
Mr Drakeford said many people in Wales already choose to wear face coverings but these will be compulsory for the first time from Monday.
“There will be signage, there will be policing, but policing is the last resort,” he told the BBC.
“If people wilfully refuse when they’ve had the rules explained to them and the reasons why we’re having to do it have been set out for them, then of course in the end you have to take action.
“But that’s the last resort and not the first resort in Wales.”
Mr Drakeford told the BBC that cases of coronavirus are growing in the country, particularly in the south-east of Wales.
A local lockdown came into force in Caerphilly county borough on Tuesday evening to control a rise in cases.
Restrictions on the 181,000 people who live there include a ban on extended households meeting indoors.
Measures are also being taken in Rhondda Cynon Taf and Merthyr Tydfil, where people have been asked to only use public transport for essential reasons and to wear face coverings indoors and at work.
“We know that that transmission is coming from people meeting people they know,” Mr Drakeford said.
“It’s not passing strangers, it’s not in the outdoors, it’s when people get into one another’s households.
“For that reason, while we continue to have an extended household arrangement where any four households can get together and create an extended household, only six people from that extended household will be able to meet indoors together at any one time.”
Enhanced public health action is being taken in Rhondda Cynon Taff and Merthyr Tydfil local authority areas to limit the spread of Coronavirus following an increase in cases.
More information here:https://t.co/Lw2mK8DfvE pic.twitter.com/ibV5sg7IDP— Public Health Wales (@PublicHealthW) September 10, 2020
The new restriction will not affect outdoor spaces, where groups of up to 30 people can meet.
“We have no evidence at all in Wales that the virus is being spread by people meeting together in the open air,” Mr Drakeford told the BBC.
“I think the evidence on this has been very clear for a long time. The virus doesn’t thrive in the outdoors and in the sunlight. It does thrive indoors and in the dark and the cold.”
Mr Drakeford is expected to announce the changes in Wales at a press conference in Cardiff on Friday afternoon.
Prime Minister Boris Johnson previously said that a limit of six people for social gatherings would come into force in England on Monday.
In Scotland, First Minister Nicola Sturgeon also confirmed that a maximum of six people from two households would be allowed to meet together.
Children under the age of 12 will not count towards the limit of six people, Ms Sturgeon said.
There are some exemptions to the new rules announced in both England and Scotland.
On Thursday, Public Health Wales said a further 102 people had tested positive for coronavirus, taking the total of confirmed cases in the country to 18,931.
Of the new cases, 34 were in Caerphilly, while nine were in Cardiff and nine in Rhondda Cynon Taf.
No further deaths were reported, with the total number of deaths since the beginning of the pandemic remaining at 1,597.
