HM Revenue and Customs (HMRC) said fines would not be enforced for anyone who files by 28 February.
The move has been met with surprise. Typically, not meeting the deadline results in an automatic £100 penalty.
HMRC said Covid had put added pressure on individuals and tax advisers to complete online submissions.
The tax agency said 6.5 million customers had already filed their tax returns for the 2020-21 financial year, just over half of the 12.2 million required to do so.
"We know the pressures individuals and businesses are again facing this year, due to the impacts of Covid-19," said Angela MacDonald, HMRC's deputy chief executive.
"Our decision to waive penalties for one month for self-assessment taxpayers will give them extra time to meet their obligations without worrying about receiving a penalty."
It is the second year in a row that such a decision has been taken on fines, owing to the pandemic.
In addition, anyone unable to pay their self-assessment tax by 31 January will not receive a late payment penalty if they pay their tax in full, or set up a time to pay arrangement (which spreads the cost over time), by 1 April.
However, interest will still accrue on any unpaid tax from 1 February.
Groups representing tax advisers and accountants welcomed the move, but said it was also designed to ease pressure on HMRC.
"It is a very surprising and unexpected move, acknowledging possible difficulties taxpayers have faced because of the recent impact of coronavirus," said Nimesh Shah, chief executive at Blick Rothenberg.
"It is a repeat of what they did last year even though the disruption from the pandemic this year has not been as profound.
"However, HMRC may be more concerned by the number of returns which remain unfiled, and the pressure to extend the timeframe as the deadline became closer."
Lucy Frazer, financial secretary to the Treasury, said: "We recognise that Omicron is putting people under pressure, so we are giving millions of people more breathing space to manage their tax affairs."