It follows a Scottish Government review of the £202 million a year scheme, which sparked concerns that eligibility could be narrowed because of its rising cost.
In fact, Mr Matheson said the national concessionary travel scheme would be extended to include those travelling with eligible disabled children under five.
He said more than 3,000 families and children could benefit.
Free bus travel for modern apprentices will also continue to be considered.
More than 1 million people benefit from the scheme, which accounts for one third of Scotland's 390m annual bus journeys.
The scheme was introduced by the Labour-Liberal Democrat coalition in 2006, with its annual cost increasing from an initial £159m.
In England, the eligible age has increased incrementally from 60 since 2010, based on the increasing female pension age, and will reach 65 in 2020.
BACKGROUND: Ministers warned not to curb 'lifeline' free bus travel schemeA review of the scheme last year found two in three of nearly 3,000 people who responded to the accompanying consultation wanted the bus pass age to remain at 60.
Mr Matheson said: “The bus pass is a benefit that many people enjoy and use as part of their daily lives and I’m delighted that this government will expand the scheme to include companions of eligible disabled children aged under five.
“The concessionary travel scheme enables independence, accessibility and social inclusivity.
"We have listened closely to the many respondents who feel that the free bus pass should remain available to all from the age of 60 and concluded that we should not change the age of eligibility.
"We will also continue to explore options to provide free bus travel for modern apprentices, while keeping the scheme under review and maintaining a balanced budget."
Kayleigh Thorpe, head of campaigns at ENABLE Scotland, said the learning disabilities charity had called for those accompanying disabled children to be included.
She said: “The cost of travel can prove a significant burden upon families who may already be under substantial financial pressure with the additional costs of raising a disabled child."
Scottish Liberal Democrat transport spokesman Mike Rumbles said: “I am very happy indeed that this scheme designed by a Liberal Democrat minister in the coalition government is not going to be altered by the SNP.
“Encouraging all those over 60 to use the bus is a win-win situation.
"Individuals benefit, our transport system benefits but above all the Environment wins as we continue to encourage people out of their cars and onto our buses.”
Scottish Labour, which also claimed credit for introducing the scheme, claimed the announcement was a "U-turn".
It said there were options to raise the age in the consultation, although there was a further option to leave it unchanged.
Connectivity spokesman Colin Smyth said: “This is a welcome U-turn from the SNP government, after months of uncertainty over the future of the bus pass.
“The free bus pass for over 60s was one of many great achievements of Scottish Labour governments.
"That is why Labour pledged to table amendments to the transport bill to enshrine the qualifying age of 60 for the bus pass for older people into law.
"It is clear the SNP would have lost a vote on this and been forced to ditch any move to increase the qualifying age to 65.
"Labour will still table the amendments and I hope now the SNP backs our plans to explicitly write into law the qualifying age of 60 to give it increased protection.
"It is disappointing the government have simply said it will just work towards free bus travel for modern apprentices, but have failed to give a clear timescale."
Scottish Greens spokesman John Finnie said: “Keeping the age of eligibility at 60 is the right thing to do, but if the government is really determined to improve bus services then it’ll get a move on and help replicate the publicly-run success story that is Lothian Buses across Scotland.”