The Fermanagh and South Tyrone MLA, who made a high-profile defection from the Ulster Unionist Party in 2004, has been endorsed by the party's deputy leader, Nigel Dodds, who ruled himself out of the running for the top job on Monday.
Mrs Foster said she had been "humbled" by the level of support.
She said: "I am fully aware of the awesome challenge and indeed great responsibility that comes from being leader of the largest unionist party and indeed the largest party in Northern Ireland but I am very much looking forward to that prospect."
Mr Robinson, 66, announced his intention to resign as DUP leader and Stormont First Minister last month, days after signing a political deal with Sinn Fein and the British and Irish Governments to stabilise the rocky power-sharing administration in Belfast.
If Mrs Foster is elected party leader there would be a degree of expectation that she would become Northern Ireland's new First Minister as well.
She added: "I do of course have still a process to go through in terms of the electoral college and the ratification by the executive of the party but if that goes according to plan I look forward to leading the party in a positive way with a positive vision for Northern Ireland in the future.
"We have much to celebrate and there is much good work to do."
Mr Dodds' decision not to run came as a major shock, as there had been widespread speculation that he would become the new party leader and would then nominate Mrs Foster as the new First Minister.
East Antrim MP Sammy Wilson had originally said he was not going to run for leader and backed Mr Dodds for the post. But, after his colleague signalled he was not going to run, he said he was going to reconsider his decision.
Mr Wilson said after reflecting on the matter he still felt it best not to stand.
"When I balanced it all up and looked at the pros and cons and the arguments for and against, I just felt it was better there shouldn't be a contest at this time," he said.
"I am willing to work with Arlene once she becomes leader."
The DUP's MLAs, MPs and sole MEP are due to meet to officially select their new leader in an east Belfast hotel next week.
Mrs Foster, who stood in as First Minister for six weeks in 2010, said her style of leadership would differ from Mr Robinson.
She said: "I am of course following not just a big act but an enormous act and I fully recognise that of course. But by the very nature of being a different person and a different gender I will have a different style there is no doubt about it."
As the deadline for nominations closed at 5pm on Wednesday, Mr Robinson took to Twitter to confirm Mrs Foster was the sole candidate.
He said: "The proposal to endorse Arlene as party leader will be put to a specially convened meeting of the DUP electoral college.
"The resolution of the electoral college shall go to the party's central executive committee for ratification immediately thereafter.
"I look forward to handing over my @DUPleader handle to the new leader.
"On 11th Jan - the first Assembly day after Christmas recess - I shall stand down as First Minister and my successor will be appointed."
If appointed First Minister, Mrs Foster, a mother of three, will be the first female leader of the devolved Stormont administration.
Any difficulties in sharing power with Sinn Fein would be "set aside" for the good of the country, she said.
"That has been a challenge and it has been a challenge since we have been in government with Sinn Fein.
"I know what it is to work alongside people with whom I have fundamental differences but I do so in the knowledge that it is good for Northern Ireland and it is good for the people of Northern Ireland and therefore any difficulty I have in relation to that I have to set them to the side to do what is right for Northern Ireland," said Mrs Foster.
Her relationship with Deputy First Minister Martin McGuinness will also come under scrutiny after her predecessor Ian Paisley and Mr McGuinness were dubbed the "chuckle brothers" while Mr Robinson and the Sinn Fein MLA were known as "the brothers grim".
She said: "He is no stranger to me.
"I have worked alongside him in the Executive and will continue to do so because I believe it is right for the good of Northern Ireland."