There was already capacity in the prison system for this increase, and the new funding would be spent on recruiting additional staff.
The main driver behind the growing population was tougher bail laws introduced by Ms Collins when she was Justice Minister in 2013.
The law changes reversed the burden of proof for accused offenders in bail cases, and was expected to lead to 350 more people a year being refused bail.
The reforms coincided with a campaign led by victims' families called "Christie's Law", which demanded even stricter bail laws.
Ms Collins said she had no regrets about her reforms and the additional costs they had created.
"I don't resile from those at all. I am fully confident that the right people are in prison."
She said the Government might have to change its forecasts to account for a higher prison population and its associated costs.
"It just means that we have to adjust to that. The fact is that the public who we represent expect that violent recidivist defenders with long track records shouldn't be getting bail when they're going to commit further crimes."
The bail changes have affected remand prisoners in particular. Remandees made up 23 per cent of of the prison population in 2014/15 - up from 20 per cent the year before.
Corrections' annual report said the higher remand numbers were creating a challenge for staff because prisoner movements and interaction had increased. Remand prisoners did not take part in rehabilitation programmes, had higher security and segregation requirements, and were over-represented in violent incidents.
When Serco lost its contract to run Mt Eden Corrections Facility earlier this month, it blamed the growth of remand inmates for its troubles at the jail.
Ms Collins took over the Corrections portfolio on Monday. The new funding was signed off by Cabinet last week, while former minister Sam Lotu-Iiga was still in charge.
Another $2.6 million had been allocated to improve the electronic monitoring of ex-prisoners in the community.
Corrections' electronic monitoring system came under greater scrutiny earlier this year after a series of escapes and crimes by people wearing electronic bracelets.
New Zealand prison population:
Sept 2010 - 8,747
Sept 2011 - 8,488
Sept 2012 - 8,623
Oct 2013 - bail laws changed
Sept 2013 - 8,474
Sept 2014 - 8,700
Nov 2015 - 9171*
*Sept figures not available for this year