The bill now goes to a select committee for public submissions.
A spokesman for Finance Minister Bill English said, if the bill passed its third reading, the Government would use its financial veto to block it, as the policy was not budgeted for.
National has threatened to use a financial veto to block the bill becoming law, saying New Zealand cannot afford such an extension to paid parental leave.
This morning Ms Moroney said that would put National on the wrong side of public opinion.
"Threatening a financial veto is an extreme measure that has never been used on an entire bill in New Zealand before.
"This is a desperate measure from John Key at a time when he's happy to spend millions on his frivolous flag fiasco and other distractions."
Ms Moroney did not know until the vote whether Mr Dunne would support it not.
He supported a similar bill sponsored by Ms Moroney in February but it was voted down by National and Act.
Since then, the Northland byelection has changed the balance of power and National now needs at least two MPs from its support partners from Act, United Future or Maori Party - to defeat any Opposition bill.
Act leader David Seymour is claiming credit for getting the Government's agreement to extend paid parental leave entitlement to parents of premature babies - also part of the Moroney bill defeated in February.
Speaking to her bill last night, Ms Moroney said: "The argument that extending paid parental leave to all for the first six months of life is a wise investment in New Zealand's future is as powerful as ever.
"Early investment and intervention is a much better option than having to spend more taxpayer money on the bad outcomes of getting it wrong."
National had appeared to change its attitude to Ms Moroney's bills, with all of its MPs last night paying tribute to her - and congratulating her for having the luck to get so many bills drawn from the private members' ballot.
Previously it has been dismissive of her bills.
Green MP Denise Roche said Ms Moroney's record in keeping the issue alive in Parliament had made National move on the issue to increase the leave -announced in last year's Budget.
Mr Dunne did not speak on the bill, the Parental Leave and Employment Protection (Six Months' Paid Leave and Work Contact Hours) Amendment Bill.
Parties that supported the first reading were Labour, Greens, New Zealand First, Maori Party and United Future. National and Act opposed it.
Paid parental leave in New Zealand began on July 1 , 2002 with 12 weeks leave.