Elizabeth Warren said she beat the odds years ago in her race for the Senate — and she could do the same thing in her campaign for president. The Massachusetts senator spoke with CBS News' Caitlin Huey-Burns a day before a new CBS News Battleground Tracker poll showed a surge in Warren's popularity, with more voters believing she could defeat President Trump.

Warren sat down with Huey-Burns in Manchester, New Hampshire, on Saturday. Huey-Burns asked Warren what she says to people who may be inclined to support her, but aren't sure if she can defeat Mr. Trump.

"Look, I know how to fight and I know how to win. I've been around this block before," Warren said.

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"When I ran for the Senate I'd never run for public office before and people said to me, 'A woman can't win this seat,'" she added. "A very competent woman had just lost it."

Warren taught law at several universities — including Harvard and the University of Pennsylvania — before launching a Senate bid for the 2012 election. She said at one point, the polls had her down by double digits. But she ended up defeating the Republican incumbent, Scott Brown, and becoming Massachusetts' first female senator. She won a second term in 2018 with 60% of the vote.

Senator Elizabeth Warren in an interview with CBS News in New Hampshire on Saturday, September 7, 2019. CBS News

"I make every day count," Warren said about her campaigning style. "And I stay in the fight for all the right reasons."

The latest CBS News/YouGov Tracker shows an increasing number of Democratic voters believe Warren could win the White House.

The survey shows that the current Democratic front-runner, former Vice President Joe Biden, receives much support for his perceived electability. About 3 in 4 voters who are considering Biden think he would beat Mr. Trump — the highest number for anyone in the Democratic race.

But Warren has made substantial gains on her electability. In the latest poll, 55% of voters think she could beat Mr. Trump — up from 39% in June.

In an aggregate vote preference across early-state contests through Super Tuesday, Warren leads the pack with 26% of voters picking her as their first choice. That's a one-point lead over Biden.

Biden's support has largely held and even risen in some places like in Iowa, so the movement toward Warren is not attrition from Biden, but of lower-tier candidates losing supporters who have gone to Warren or, in lesser numbers, to Sanders.

Meanwhile, Warren leads the entire Democratic field for certain candidate qualities. A majority of voters — 42% — said in the survey that she is the most knowledgeable about the issues in the 2020 race. Warren also scored the highest for enthusiasm, with 46% saying they'd be enthusiastic if she became the nominee, compared to 29% for Biden.

Biden still clings to a narrow lead over Warren for convention delegates, which will ultimately decide the Democratic nominee. Biden has an estimated 600 delegates available to him through Super Tuesday, compared to 545 for Warren.

Warren said she attributes her rise to "what I fight for and how I fight." She said Americans are getting frustrated with inaction on issues like gun safety, prescription drug prices and climate change.

But Warren shook her head when Huey-Burns asked if she ever imagined she would be a front-runner.

"I think of this as, I'm in it with more and more and more people," Warren said. "We're doing this together, side by side."

Anthony Salvanto contributed to this report.