The measures in the bill introduced yesterday were designed to save $138 million over four years.
Mr Power said the changes would not affect cases involving vulnerable parties, care and protection of children, and serious criminal matters.
Among its measures, the bill will limit access to legal aid for people facing "less serious criminal cases" to those earning less than $22,000 for a single person or $50,934 for an adult with two dependent children.
Those earning above the threshold would still be able to apply for legal aid but would have to prove that their case was expensive or that they were unable to pay for a lawyer.
But Labour's justice spokesman Charles Chauvel said extending the means thresholds for criminal charges was disturbing.
"If $22,000 is the new definition of rich then I think we should all be pretty concerned. It shows the Government is out of touch with what ordinary people need in terms of getting a hand to get access to justice when they need it."
The thresholds made more sense in civil cases in which participation was often voluntary, but criminal cases, even "less serious ones" were a different matter as they could lead to unfair or unsafe convictions. .
"Times are tough but rather than looking at savage entitlement cuts, what the Government could do is look at the possibilities of making legal aid more of a loan than a grant.
"It would be good to see more effort being put into things at that end rather than saying we're going to cut access to justice off at the front end instead."