Dixon said voters "are fed up with the government".
In her victory speech, she said they had delivered a message to Sunak that the Conservatives "no longer have a mandate to govern".
"I don't think they believe that the Conservatives have the answers, I think they think it's Labour's turn now," she added.
Lord Hayward, a Conservative peer and elections analyst, said the election went "generally as expected, no shocks but definite satisfaction for the Labour Party".
He told Sky News that there is "no question" the Tories have a "real challenge on their hands".
The contest was triggered by the resignation of previous Labour MP Christian Matheson.
He stood down after Parliament's watchdog recommended his suspension for "serious sexual misconduct".
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