Labour says the 40-minute wait to get through to the 111 emergency number at the weekend could have been the difference between life and death.
Police are blaming a technical fault for the system shut down from midnight Saturday to 12:45am on Sunday which meant callers heard an automated message saying the system was overloaded.
A back-up system was used and it took between 35 to 40 minutes to clear the backlog of calls.
An investigation is now underway into the nationwide fault.
Labour's police spokesman Stuart Nash says the minister Judith Collins must ensure the problem never happens again.
"The Police have said the fault lies with their system. This must be fixed immediately. There cannot be a repeat this Saturday, it's the peak time for emergency calls. Just telling people to keep calling back isn't the solution.
"People don't call 111 for chat. They are in dire need of help. This isn't a glitch. A 40-minute backlog of calls is the difference between life and death."
Ms Collins' office says they won't comment until police have finished their investigation.
Mr Nash says underfunding police for years like the Government has done can lead to issues such as this.
Inspector Mal Schwartfeger says the fault was with the police system rather than telecommunications company Spark. Spark found the problem and told police.
The main system was restored by 2am on Sunday.
He apologised to the public and said police were taking the necessary steps to ensure the fault doesn't happen again.