"When James went to meet President Obama, President Obama ran across the room, embraced him and thanked him profusely for his time, by the way," Carter said.
The video, which shows Romney describing nearly half of Americans as "dependent" on government, created a national controversy and was thought to deal a serious blow to the Republican nominee's presidential chances.
"There are 47% of people who are with him [Obama], who are dependent on government, who believe that they are victims, who believe the government has to care for them, who believe that they are entitled to health care, to food, to housing, you-name-it," Romney said at a private Florida fundraiser last year.
James Carter, the president's grandson, reportedly encouraged the owner of the video to turn it over to Mother Jones magazine.
In the interview airing Thursday, President Carter said the release of the tape was a "pivotal moment" in the contest.
"It was something [Romney] could not deny and it stuck with him for the rest of the election and I think it was a major factor, if not the major factor," he said, adding he believed it won Obama the election.
Obama met with Carter's grandson while in Atlanta for a post-State of the Union rally to champion his universal pre-K initiative.