Axelrod predicted that the tax compromise will be able to pass before the end of the year.
But some Democrats have been so fierce in their criticism of Mr. Obama's willingness to let Republicans extend tax cuts for the highest income levels that they've suggested he's betraying the party.
"Democrats are so mad that the President may wind up with an opponent in some of the early primaries in 2012. Do you worry that President Obama might wind up being a one-term president?" asked CBS chief Washington correspondent Bob Schieffer.
"No, I don't worry about that at all, Bob," Axelrod replied. "The thing that would be worrisome to me is if we made a bunch of decisions based on short-term political calculations and forgot about why people sent us here."
Earlier in the week Sen. Bernie Sanders, D-Ver., spent more than eight hours on the Senate floor, which he criticized as a giveaway for "millionaires and billionaires who don't need it."
The Bush tax cuts are set to expire on January 1. Democrats want the tax cuts maintained for the lower and middle class families and small businesses, but Republicans want to extend the tax cuts no income limits, including the super rich.
Many liberal Democrats in the House are up in arms about what they call a Republican "blackmail."
But the president said it is imperative to keep the tax rate for the middle class, and heto urge Democrats to accept the compromise.
Many House Democrats want to scale back a provision that would allow estates as large as $5 million to escape taxation.
The Senate could vote on the tax plan Monday, and Axelrod predicted that the House will get to it later in the week.
"Right now we're facing a deadline - we ought to get this done," Axelrod said. "I believe that this will pass."
"I understand again the objections that people have in our party to some elements of this compromise. But I say again that compromise necessitates accepting some things that you don't like. That's the nature of compromise," he said.
Howard Dean, the former chairman of the Democratic National Committee, another fierce opponent of the tax compromise, nevertheless said that having an opponent go up against Obama in the Democratic primary would not be a good thing.
"I don't think he's going to face an opponent in the Democratic primary. I think that would be a bad thing for the country. I think it would be a bad thing for the Democratic Party," Dean, who was also on Sunday's Face the Nation, said.
"The history of people running against presidents in their own party is the challenger loses and then the president is weakened and loses. The president has done some things that I think are terrific. This is not one of them. But I think he will not get an opponent."
Dean and other liberal Democrats want the tax cut bill rewritten, but Axelrod said he doesn't see any major changes happening to the compromise plan, despite their objections.
"We demanded that we extend the child tax credit, the college tax credit, the earned income tax credit, and we got these additional tax cuts that are going to be very helpful to people across this country," Axelrod said. "I think the game work in place is probably the one we're going to vote on."
Axelrod said the important thing was to get down to business now, and to worry later about whether President Obama was losing support.
"What we want to make sure is that we don't get into a kind of Washington-style gridlock where this flows over into the New Year and we send our economy in the wrong direction," Axelrod said.
"In this case it's absolutely necessary that we get this done. The president has done the right thing by focusing on what's important. You know, the rest of the politics will take care of itself down the line."
Schieffer asked Axelrod whether he was concerned about the level of opposition against the president within his own party. "They're mad about it. They're in a rage. I mean, somebody in the house using the F-word to describe the president," Schieffer pointed out. "Some of this language they're using to describe the president [is] telling him basically to shove it."
Axelrod was unperturbed. "You know what? The president of the United States has a bigger concern," Axelrod said.