The changes, which were unveiled earlier this month, are to be debated by the Scottish Parliament.
Ahead of the debate, Mr Swinney said: “Government alone cannot realise our ambition of creating a world-class education system for Scotland. That is why I am reaching out to everyone in Parliament.
“The detail of our reforms will need to be developed in close collaboration with local government, the teaching profession, parents and young people.
“I want everyone to join with us in closing the attainment gap and raising the bar for all.”
The changes will see headteachers given a raft of new powers as they become responsible for raising attainment and closing the gap between the poorest and richest pupils.
Heads will also have the power to choose staff and management structures, decide on the curriculum - within a broad national framework - and directly control ‘’significantly’’ more funding.
The reforms, which follow a review of school governance, will also give parents a stronger voice in the running of schools, Mr Swinney said.
Some of the reforms will be contained in an Education Governance Bill, to be introduced to Holyrood next year, while others can be implemented without legislation.
Mr Swinney said: “The reforms I have announced are driven by this government’s unwavering focus on improving Scottish education so that all our young people can fulfil their potential.
“At their heart is a simple idea - we are putting more power and money in the hands of schools and teachers, who are best-placed to make decisions about the learning of young people. International evidence shows this approach works.”
Conservative education spokesman Liz Smith welcomed greater devolution of power to teachers but said more radical reform is needed.
Labour’s Iain Gray said the “first reform we need is more teachers, properly paid, properly supported and properly resourced”.