He said the money was coming from existing funds.
Health unions have been striking over pay parity and staffing levels, which they say are unsafe.
The Royal College of Nursing, which has engaged in strike action for the first time in its history, is planning to take further strike action later this month and up until March.
The GMB union has also confirmed it plans to carry out further strike action in January.
In a bid to avert the forthcoming action, Mr Swann, addressing the Assembly, said a new offer has been made to reinstate pay parity with England and that “decisive action” has been taken on staffing levels.
He said the department has gathered together £79m to increase pay for health staff to give them the same wage as workers in England and an extra £30m would be pulled from existing finances at Stormont.
Two previous offers had been made by the department to end the dispute but both were rejected by trade unions. In December the department offered a 3.1% increase in overall pay and an additional £38m was made available to the health service.
Mr Swann described a meeting with unions as “constructive”. It is understood the unions have taken the offer to consult with their members to determine what action to take.
The health minister said his department is providing a written commitment for immediate high-level engagement with unions to produce a costed implementation plan on safe staffing within an agreed short period.
He said: “The breakthrough we all wanted has been achieved. This is a good day after some very difficult days.
“I am grateful to my colleagues around the Executive table for helping to make it happen.
“We have moved significantly, and quickly, to take action- together. That is a sign of optimism for the future.
“I know there is scepticism in many quarters about what this health minister and this Executive can do for the health and social care service. That is entirely understandable.
“Many good people doubt whether we can set party politics aside and work together constructively. We shall see.
“But maybe- just maybe- today will give the sceptics some pause for thought.”
Mr Swann said that “sustained additional funding” is essential but that there is no quick fix to the crisis in the health service.
He said the “component parts” are now in place to settle the industrial dispute and that additional funding has been secured to ensure pay parity.
“I met immediately with trade union officials and briefed them on these latest developments,” he said.
“The new offer was outlined by me to Trade Union representatives this morning and will be formally submitted to them this afternoon.
“Officials will meet with trade unions tomorrow to agree on the details.”
In a joint statement health unions said they had a “meaningful briefing” with the Mr Swann and that they will consider the detail of the offer in the coming days.However, ahead of the announcement in the Assembly, the GMB trade union said its members will strike on January 24.
This will be followed by action short of strike the following day.
Jim Donley, GMB organiser, said: “It’s with a heavy heart our members go on strike – but they are angry about they way they have been treated.
“It’s now time for the politicians to deliver on their promises.
“They must step up to the plate to end the unfairness in pay and staff shortages across all disciplines.”
The RCN has announced plans to strike again on January 20, 22 and 24, as well as on February 10, 12, 14 and March 2, 4 and 6.