"It takes a big man to know when he's wrong and admit it," said Pence, adding, "Donald Trump last night showed that he's a big man."
Pence also brought up his Christian faith in his explanation of why he continues to stand by Trump, saying he believes in "grace" and "forgiveness."
In a round of television interviews earlier in the day, Pence said he never considered leaving the Republican ticket and sought to play down differences that have emerged between him and Trump on foreign policy.
"It's absolutely false to suggest that at any point in time we considered dropping off this ticket," he said on CNN. Pence said in the same interview that he believes Trump was "able to move beyond" the controversy over his 2005 comments during the debate.
Pence said on Fox News Channel that Trump "expressed genuine contrition for the words he used on the video that became public." He said he was "proud" of Trump's performance, echoing a tweet he wrote Sunday night.
On Saturday, Pence did not appear for Trump at a campaign stop in Wisconsin. He issued a written statement saying: “As a husband and father, I was offended by the words and actions described by Donald Trump in the eleven-year-old video released yesterday. I do not condone his remarks and cannot defend them. I am grateful that he has expressed remorse and apologized to the American people.” He kept quiet for most of the weekend.
But now, Pence said on CNN, he is looking forward to "campaigning shoulder to shoulder" with the real estate businessman.
In Sunday's debate, Trump was asked what he would do in Syria and specifically how he would seek to resolve the humanitarian crisis in Aleppo. Moderator Martha Raddatz of ABC News reminded Trump that Pence said in the vice-presidential debate that the United States "should be prepared to use military force” against the Bashar al-Assad regime. Trump responded: "He and I haven't spoken, and I disagree."
Pence struggled to explain that he was not odds with Trump and accused Raddatz of conflating his positions.
"Well, he said we hadn't spoken specifically about what she said. And the reason for that was because I didn't say that," Pence said in the CNN interview.
Here is Pence's comment on the use of U.S. military force in Syria from the vice-presidential debate transcript:
I just have to tell you that the provocations by Russia need to be met with American strength. And if Russia chooses to be involved and continue, I should say, to be involved in this barbaric attack on civilians in Aleppo, the United States of America should be prepared to use military force to strike military targets of the Assad regime to prevent them from this humanitarian crisis that is taking place in Aleppo.
This post has been updated.