George Osborne toured the NGI with Nobel Laureate Professor Sir Kostya Novoselov, who discovered the material.
Graphene consists of single atoms arranged in a honeycomb structure. It has been called a "wonder material" because of its potential uses.
It was first isolated by Sir Kostya and Sir Andre Geim in 2004.
Mr Osborne said the centre would "bring together leading academics, scientists and business leaders to help develop the applications of tomorrow, putting the UK in pole position to lead the world in graphene technology".
More than 35 companies worldwide have already partnered with the university to develop graphene projects.
The university's president and vice-chancellor Prof Dame Nancy Rothwell said the "state-of-the-art institute is an incredible asset, not only to this university and to Manchester, but also to the UK".
The government provided £38m for the construction of the institute via the Engineering and Physical Sciences Research Council, with the remaining £23m from the European Regional Development Fund.
The Chancellor of the Exchequer has officially opened the £61m National Graphene Institute (NGI) at the University of Manchester.