In a widely expected announcement, Mr Robinson, 66, said he will not contest next May's Assembly election and is likely to leave his post at the head of the powersharing coalition in the coming weeks.
The move comes days after he signed a political deal with Sinn Fein and the UK and Irish governments to avert the collapse of the administration.
Mr McGuinness said Mr Robinson had informed him of his intentions well before it was announced publicly.
"I have always given credit to Peter for recognising that the only way forward in this country was for us to work together," said the Sinn Fein veteran.
The warm relationship Mr McGuinness struck up with the late Ian Paisley in their time together at the head of the Stormont Executive has been well documented - in fact their unlikely friendship is the subject of a forthcoming feature film.
While Mr Robinson and Mr McGuinness's seven years in office has not been characterised with the same bonhomie, the republican insisted he has developed a strong relationship with his unionist partner in government.
"I think he was a very strong friend, like Ian Paisley, of the peace process," he said.
"And Ian paisley and I, incredibly, developed a friendship which existed until the day he died. So I do regard Peter Robinson as a friend, yes."
Mr Robinson said he had wanted to stabilise the powersharing administration in Belfast before stepping aside.
The experienced politician suffered a heart attack earlier this year but has insisted he had made his mind up to leave before the health scare.
There had been growing speculation Mr Robinson would outline his departure plans at the DUP's annual conference this weekend.
In the event, he confirmed his exit in a pre-conference interview with the Belfast Telegraph.
"I think it would be disrespectful to the party membership if I was to go through a conference with the pretence that I would be leading the party into the next election," he said.
"I think they have a right to know what the circumstances are."
North Belfast MP Nigel Dodds will be among the favourites to take over as DUP leader.
However, with Mr Dodds based in Westminster, another senior party figure could take on the role of Stormont First Minister.
Northern Ireland's finance minister Arlene Foster has been touted as a potential leader of the powersharing coalition.
Mr Robinson said he would remain in the post until Tuesday's Fresh Start agreement is "bedded in" - a period he indicated could last into the early new year.
The East Belfast Assembly member, who replaced Ian Paisley as first minister and DUP leader in 2008, said he had wanted to secure a number of specific objectives before leaving - namely stabilising the powersharing government, the DUP retaking the East Belfast Westminster seat he lost in 2010 and setting a date for Northern Ireland to determine its own corporation tax rate.
With all those accomplished, he said the time was right to step aside.
"For anyone who is not very young to go beyond two terms is stretching it," he said.
"There are massive pressures on anybody in this job. You do need to renew political leadership, bringing in people with perhaps more energy and people with new ideas."
Tuesday's Fresh Start agreement resolved the wrangle over the non-implementation of the UK Government's welfare reforms, and a number of other disputes which had pushed the coalition Executive to the verge of collapse, including the fall-out from a murder linked to the Provisional IRA and an acute budgetary crisis.
However, the accord has been fiercely criticised by victims' campaigners for failing to secure consensus on new mechanisms to address the legacy of the Troubles.
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