The pair met the media on Tuesday morning along with new owner Ann Budge to explain how the new management set-up will work and some of the decisions of a dramatic and ruthless first day in office.
Levein has been installed to oversee football matters - with his first major decision to dispense with the services of manager Gary Locke for next season's Scottish Championship campaign.
But he will allow Neilson ultimate responsibility in team selection matters.
Neilson said: "I will pick the team. When Craig spoke to me on Sunday night that was one of the questions I asked him and he assured me I would pick the team. That's the reason I'm sitting here."
Levein revealed he had first met Budge before agreeing to endorse the Foundation of Hearts' takeover attempts in August last year.
The pair continued meeting after Levein offered to help Budge plan for her assuming control - in lieu of a fans-led takeover within five years.
The talks led to a job offer and Levein admitted the decision not to renew Locke's contract - and install Neilson in a role with the emphasis on coaching and "coach education" - was made a "while" ago.
Levein said: "Trust me, yesterday was a hugely difficult day. I have got a lot of respect for Gary Locke and a lot of time for him as a person. The last thing I wanted to do was tell Gary Locke there wasn't a role for him in this new structure. But ultimately that is the fact."
The changes on Monday also saw five first-team players released, including player of the year Jamie MacDonald, while Ryan Stevenson's departure is expected to be confirmed after the club invoked a clause to reduce his wages following their relegation.