Komen said it would change the criteria "to make clear that disqualifying investigations must be criminal and conclusive in nature and not political."
"We will continue to fund existing grants, including those of Planned Parenthood, and preserve their eligibility to apply for future grants," the statement said.
Debbie Mintowt, executive director of the Komen affiliate in Utah, said Friday morning she can only assume the national organization "listened to the nationwide dialogue," since its announcement earlier this week.
Her Salt Lake City office had heard from people both pleased and displeased by Komen's earlier announcement. Mintowt expressed hope that the politics of abortion will not affect the work of the charity.
"We really are not a political organization. We focus on breast health," said Mintowt.
"I'm thrilled," said Karrie Galloway, director for Planned Parenthood in Utah, upon learning of Komen's reversal. Breast cancer awareness and research, she said, "was the worst possible thing to put politics into."
She added: "I'm sorry, but Komen set themselves up for this. I don't think they realized how women and the people who love them don't like feeling they were hoodwinked."
Still, she said, "I hope everyone can let the past few days go and we'll start anew."
Planned Parenthood in Utah has not received Komen funding since 2008, when it used a grant of $11,925 to create laminated breast-examination guides to hang in showers.
Many of Komen's affiliates across the country had openly rebelled against the decision to cut the funding, which totaled $680,000 in 2011. One affiliate, in Aspen, Colo., had announced Thursday that it would defy the new rules and continue grants to its local Planned Parenthood partner.
In addition, Komen was inundated with negative comments via emails, on Twitter and on its Facebook page. Many of the messages conveyed a determination to halt gifts to Komen organizer of the popular Race for the Cure events because of the decision.
Meanwhile, Planned Parenthood was reporting an outpouring of support donations large and small, triggered by the Komen decision, that it said surpassed $900,000.
Planned Parenthood's president, Cecile Richards, thanked those donors Friday and welcomed Komen's change of heart.
"We are enormously grateful that the Komen Foundation has clarified its grantmaking criteria," Richards said. "What these past few days have demonstrated is the deep resolve all Americans share in the fight against cancer."
Through the Komen grants, Planned Parenthood says its health centers provided nearly 170,000 clinical breast exams and more than 6,400 mammogram referrals over the past five years.
Komen, in its statement, said it was immediately starting an outreach to its affiliates and supporters to get the charity back on track.
"We urge everyone who has participated in this conversation across the country over the last few days to help us move past this issue," Komen said. "We do not want our mission marred or affected by politics anyone's politics."