The infrastructure organisations will work with Candour Collaborations, a consultancy, to deliver 57 workshops in England and Wales to help third sector organisations with aspects of contract bids.
The workshops will help charities to apply for, negotiate and deliver rehabilitation contracts and will include sessions on measuring the cost of delivery and demonstrating the impact of their work.
The partnership is aiming to have about 4,600 organisations attend workshops. Sessions will start this summer.
The MoJ will begin signing contracts with bidders from next year and contracts will go live in autumn 2014.
The government grant is part of a £500,000 pot designed to help the third sector bid for rehabilitation contracts.
A spokeswoman for Acevo said her organisation was keen to avoid a repeat of the mistakes made in the Work Programme, in which the voluntary sector played "a marginal role" and voluntary providers tried to deliver contracts that were not commercially viable.
She said Acevo was pleased that its bid to strengthen the third sector’s role in delivering the MoJ contracts had been successful.
"The MoJ has taken on board many of our ideas about working with the sector to use our skills and expertise in working with the hard end of offending and tackling the revolving door that sees so many released prisoners back in jail within a year," said Sir Stephen Bubb, chief executive of Acevo.
The NCVO said offenders and the wider community would benefit from the involvement of expert voluntary organisations in the rehabilitation programme.
"Clearly, there are still challenges to come in the rehabilitation reform programme, but I hope we can give voluntary organisations the skills and confidence to fight for the best deal from private contractors and commissioners," said Karl Wilding, director of public policy at the NCVO.
The MoJ said it made the grant available so that the voluntary sector could play a full role in its "essential reforms" of offender rehabilitation.