The Local Government Association called for a raft of measures to accompany the legislation ahead of it receiving its second reading scheduled this afternoon.
It came two weeks after the Supreme Court ruling that Prime Minister Boris Johnson's suspension of parliament was unlawful, restoring the bill to the Commons agenda after it was temporarily dropped.
The draft legislation, first published in January and given its first reading in July, is intended to tackle rises over the past decade, in domestic abuse reports, referrals to children's social care services and in requests for refuge placements.
Research suggests up to 800,000 children live in homes where domestic abuse occurs, with the risk of lasting emotional damage to them.
The bill includes the first statutory definition of domestic abuse, which Justice Secretary Robert Buckland told MPs would help drive public bodies' understanding of the issue and be publicised at schools.
It includes the appointment of a domestic abuse commissioner - named last month as Nicole Jacobs, former chief executive of charity Standing Together Against Domestic Violence.
The debate featured the first speech by former PM Theresa May since she returned to the backbenches.
May, who played a key role in her government's domestic violence strategy, lamented that so many people do not know a "good relationship", and highlighted the impact on children and other family members.
May said the potentially landmark legislation had the power to change society's attitudes to domestic violence.
Ahead of the second reading, the LGA said: "In order for the bill to have real success in tackling domestic abuse and creating consistency of services, it must be underpinned by adequate, long-term funding in key services including children's services and housing."
The association also suggests a cross-government approach to the problem, incorporating health, housing and education, with an equal focus and funding for prevention of domestic violence and early intervention.
"While it is right that we should prioritise and support victims, breaking the cycle of domestic abuse will also mean stopping perpetrators from reoffending," said the LGA.
"This requires funding and investment to be put towards evidence-based perpetrator programmes."
The association also calls for the key learning and best practice from domestic homicide reviews to be shared on a national level, to contribute towards the commissioner's annual report.
It adds: "With domestic abuse a factor in the majority of child protection cases, we would like to see more emphasis on how children can be supported when they have experienced domestic abuse.
"There must be greater investment in child and adolescent mental health services and early intervention work to mitigate the impact of adverse childhood experiences based around domestic abuse.
"This legislation comes at a time when local government, and particularly children's services, are facing unprecedented demand.
"Councils have worked hard to protect budgets for essential child protection services, but funding pressures have led to difficult decisions in other parts of the service, reducing vital early intervention work and leaving children and young people unable to access support until they reach breaking point."
Children's charities welcomed the progress of the bill and praised MPs for highlighting the plight of children in abusive relationships.
Great to hear many MPs raising concerns around support for children who experience domestic abuse in the #DomesticAbuseBill— AfC Policy (@AfC_Policy) October 2, 2019
Very happy to see #domesticabusebill having its second reading today, hugely important legislation which we hope will be improved throughout its stages to better protect and safeguard children and young people experiencing domestic abuse.— The Children's Society Policy Team (@ChildSocPol) October 2, 2019
Absolutely right that children are affected even if they are not in the room, often more terrifying hearing abuse, powerless to do anything about it. Glad to hear @theresa_may call on Govt to look closely at children as victims of abuse #DomesticAbuseBill— Jessica Asato (@Jessica_Asato) October 2, 2019
There were also calls for children to be included in the legal definition - an omission that the NSPCC has previously highlighted.
Imran Hussain, director of policy and campaigns at Action for Children, welcomed the bill as an opportunity to support children but echoed these concerns.
"Far from having the safe and happy childhood they need, every day our frontline workers see the emotional scars of domestic abuse on children - from nightmares, flashbacks and bed-wetting to depression, or even wanting to end their lives," said Hussain.
"If the bill is to deliver what's required for these children, then it must recognise them as victims, not just witnesses, and ensure specialist services are available to all children who need them.
"We and other charities, along with survivors, have worked closely for nearly two years with officials and ministers to strengthen the bill, it's now time for the government to deliver."
Buckland sought to reassure MPs that the legislation would be carried over to the next session if parliament is suspended again for a Queen's Speech, after concerns were raised over the timing.
Campaigners welcomed Johnson's September promise to resurrect the lost legislation in the Queen's Speech - he tweeted at the time that the government was "fully committed to tackling the issue".
Domestic abuse shatters lives & tears families apart. We are fully committed to tackling this horrific crime - which is why the Queen's Speech will confirm we will be reintroducing domestic abuse legislation in the next session.— Boris Johnson (@BorisJohnson) September 12, 2019