Ahead of the showdown Mr Smith said that he was the only candidate in the leadership race who could heal the divides within Labour and take the fight “back to the Tories”.
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The head-to-head came just weeks before voting in the race closes, with the victor to be named on September 24.
The two also clashed over the party’s record on tackling anti-Semitism and abuse within Labour, with Smith questioning whether Corbyn was “entirely committed” to it.
Mr Smith said he admired Mr Corbyn’s “optimism” but added: “I don’t think that’s what we will see”.
“Jeremy Corbyn says he condemns attacks on Labour Party staff, saying they should not be used as a “political football”.
During the debate, Corbyn said MPs would come together to oppose the Conservative government following the leadership contest.
In a special edition of BBC1’s Question Time, the Labour leader said he had been talking to MPs who were willing to get behind him despite the mass resignations of shadow cabinet members, citing a lack of confidence in Corbyn’s leadership, that triggered the contest.
A Jeremy For Labour spokeswoman said: “Smith’s dodgy dossier is a new low in his increasingly desperate and negative campaign”.
Mr Smith told the audience he was “incredibly confident” that he would be victorious on September 24.
His comments come after his aides suggested on Wednesday that he might rule out full membership of the single market unless the United Kingdom could negotiate exemptions from key European Union rules.
Mr Smith said he wants Labour to promise at the 2020 general election to take Britain back into the EU. Corbyn said he wanted to remain in the single market “if it’s possible, and I think it probably is”.
Asked if that means to ignore the Brexit vote, he replied: “Well, exactly”.
Mr Smith claimed “hard-left” activists were “flooding” in to the party to support Mr Corbyn.
Jeremy Corbyn and Owen Smith have been told that Labour’s bitter election contest has made them both look “unelectable” and they should step aside for another candidate, as the pair clashed in a series of tetchy exchanges. “I have spent my life opposing racism in any form, as have you”, Corbyn said, making clear he had never supported Hamas.
The Strictly star and former Labour cabinet minister said the hapless leftie is content for the party to just grow its membership in opposition.
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Mr Corbyn’s team described it as a “dodgy” collection of rehashed claims that shows Mr Smith’s desperation.
