The accusations against Northumbria Police were made on the first day of a High Court civil claim by his sister Debbie Essery and brother Darren.
It will focus on the crucial minutes between Moat dialling 999 to say he was hunting for police and him shooting the unarmed traffic officer who was sitting in his patrol car on a prominent Newcastle roundabout above the A1.
The claim states that had he been warned about the specific threat by the force, Pc Rathband would have kept mobile.
In the minutes after he was blasted twice by the ex-doorman in July 2010, senior officers ordered all unarmed police to return to their stations.
Geoffrey Tattersall QC, representing Ms Essery and MrRathband, said they were "dealing with a very short space of time" but that David Rathband had been left ignorant to the threats that Moat had made against officers after he had shot his former partner and her new boyfriend, killing the latter.
Addressing Mr Justice Males at Newcastle Crown Court, he said even though Moat had rung 999 and made threats to the police, "no warning of any kind was issued. David Rathbandwas shot at a time when he remained ignorant to the threats made by Raoul Moat to officers".
He said: "Moat had said if the boyfriend had not been a police officer he would not have shot him. This fact is relevant in any assessment of the risk he posed.
"A duty of care was owed and the force was negligent in failing to warn David Rathband that Raoul Moat had telephoned and made threats.
"We say that an interim warning should have been given noting the threat to officers, to probably tell them to be vigilant and probably that they should keep on the move on the basis a mobile officer is far harder to shoot."
He said that Northumbria Police went on to produce new guidelines in December 2010 for when threats were made against officers. These included the instruction that information concerning the threat is broadcast at the earliest opportunity, that officers must remain vigilant and that they should avoid static positions.
"A plain reflection of what should have happened that night," he said. "It was a complete no brainier to warn of the potential risk and it should have been given from the outset.
"I do criticise the police for failing to give that interim warning which we say would have made a difference."
Speaking outside court before the case began, Ms Essery said she and David's twin Darren were entrusted by their brother to continue the litigation against Northumbria Police.
She said: "David felt he was left out in harm's way by the organisation in which he served, the organisation which has continued to state that they did nothing wrong, however have since changed various practices and procedures to ensure it never happens again."
She said should the claim be successful, neither she nor Darren will benefit financially, and the money will go to David's children Ash and Mia.
And she paid tribute to Pc Rathband, who killed himself in February 2012, saying: "David was a much-loved son, brother and uncle whose passing has left a huge void in our lives and he will always be sadly and deeply missed by his family."
Also in court was Mr Rathband's former wife Kath, who briefly answered questions in the witness box.
She said that former Chief Constable Sue Sim had told David she would not hold it against him if he took action against the force for what happened.
The case continues.