The law covers both penalties against owners who lose control of their dogs and those who don't look after their canines properly.
Some of the offences that will now bypass the court system include failing to properly feed and care for a dog, not micro-chipping or registering the animal and the fraudulent sale of dangerous dogs.
More serious offences potentially carrying terms of imprisonment, as well as a few others that have been excluded, will still go through the courts.
The legislation received the backing of National, Labour and New Zealand First, but was opposed by the Greens and ACT.
Green Party co-leader Marama Davidson told the House the bill would weaken due legal process and would mean that some of the decisions being made by the court were too big to pass on.
"We do actually respect the sponsor of the bill and the problem that this legislation is trying to address," she said.
"[But] the hypothetical situation where we are dealing with an infringement with say, a $3000 fine, up to the maximum, and where dog destruction is imminent is a big deal. That is a big decision to be made."