"In fact, last week's March quarter inflation release marked the first time in over five years that headline inflation printed above the midpoint of the RBNZ's 1-3 per cent target range."
The bank's economists noted that inflation expectations were also lifting in response to higher costs.
"Firms' pricing and wage setting decisions all now have a materially higher inflation benchmark to build in. In other words, inflation is likely to beget inflation, at least over the next year or so."
They expect annual inflation to stay around 1.5 per cent to 2 per cent over the next few years and predict the Reserve Bank will keep the cash rate on hold until late 2018.
Kiwibank chief economist Zoe Wallis also said in a note that there was now much less risk of inflation expectations falling - one of the Reserve Bank's biggest concerns when it started cutting at the start of last year.
"In fact, inflation expectations are likely to move higher in response to stronger headline inflation."
Wallis said higher inflation may also result in employees demanding higher wages.
"Employees may well start demanding wage hikes in order to compensate for the higher cost of living, which in turn can push domestic prices higher."
Kiwibank said the inflation data had brought forward its expectations of cash rate increase from 2019 to 2018.
"We now expect the RBNZ to start lifting the OCR toward the end of 2018, around six months earlier than we previously expected. This is also six months earlier than the RBNZ had signalled in the February MPS [Monetary Policy Statement]."
Wallis said the bank saw limited risk of inflation overshooting the target range over the next few years.
"We expect inflation to remain well contained around 2 per cent and the RBNZ to look through temporary forces pushing inflation around."
She said there were also risks if the Reserve Bank started lifting the rate too soon, given New Zealand's current stage in the economic cycle and the indebtedness of New Zealand households.
The bank noted there had already been some de facto tightening in monetary policy, with retail deposit and mortgage rates moving higher in response to higher wholesale funding costs.
"Overall the RBNZ still has time to sit on its hands and ensure the economy continues to experience solid growth - just a little less time than we previously thought."