She said she had listened to the concerns of local authorities and the funding would allow them to deal with "the most pressing issues they face".
But Scottish Tory finance spokeswoman Liz Smith said it did not “even come close to what councils need just to maintain basic services”.
She said: “Local authorities are still facing an enormous £250m real-terms cut in April, despite a record block grant from the UK Government last year.
“It means Scotland’s 32 councils will now have to choose between delivering essential public services on the cheap or punishing local residents with drastic council tax hikes.
“The SNP has neglected local councils for years, and they can no longer cope with these devastating cuts.”
Scottish Liberal Democrat leader Alex Cole-Hamilton warned that “many people are feeling the squeeze like never before” because of inflation, and said: “We need to reflect that crushing reality in this Budget that we pass.
“But instead, we see cuts – and they are cuts – to local government, and they will see an unavoidable rise in council tax which will compound that reality still further.”
Scottish Labour finance spokesman Daniel Johnson said: “The new funding for councils is welcome but will not be nearly enough for our hard-pressed local councils who have been hit by over a decade of SNP cuts.”
A spokesman for Cosla, the council umbrella body, said local authority leaders will meet on Friday morning to discuss the issue.
It previously warned councils face a real-terms cut in core funding of £371m.
The independent Scottish Parliament Information Centre found the Budget would represent a £284 million real-terms cut to councils’ core budgets.
Local authorities are to be given full control over council tax rises next year for the first time since the SNP came to power in 2007.
Ms Forbes made the latest announcement during stage one of the Scottish Budget's passage through Holyrood, where it passed its first parliamentary hurdle by 69 votes to 54.
She said she was "conscious" of the challenges facing councils, including the increasing impact of inflation.
She said: "I've repeatedly said that next year's Scottish Budget is fully allocated. That remains the case.
"However, I've also been clear that I've been monitoring this year's Budget very carefully.”
Ms Forbes said UK ministers had “spent weeks advising us that we should not expect further funding”.
But this position changed in the last few days, she said, and the UK Government “has advised, in a matter of days, that we should anticipate further funding for this year”.
She said: "In light of new information from the UK Government, I now have some new and additional flexibility on this year's funding.
"And so I am pleased to confirm my intention to utilise the Scotland Reserve to carry forward sufficient funding from this year to next year, to allocate a further £120m of resource to local government.
"Councils will have complete flexibility to allocate that additional funding as they wish next year."
Ms Forbes said councils had asked for an additional £100m, adding: "We have heard them and listened and we are going to go further.
"That will allow them to deal with the most pressing issues they face and, at a time when people are understandably worried about the cost of living, I would point out that this increase in funding would be equivalent to a 4 per cent increase in council tax next year.
"So while councils have full flexibility in setting local council tax rates, I don't believe that there is now a requirement for any inflation-busting increases next year."
It came as a row broke out after it emerged an 11-strong team of Scottish Government employees, comprising a senior civil servant and ten officials, are coordinating work on a new prospectus for independence.
A Freedom of Information request revealed their combined salaries come to around £700,000 a year.
Pamela Nash, chief executive of pro-UK campaign group Scotland in Union, branded it “an obscene waste of public money”.
She said: “Rather than focusing on how to divide Scotland once again, the government’s priority should be bringing communities together, giving our NHS and public services the resources they need.”