Mike Birtwistle and Sarah Winstone quit MHP in August after 13 and seven years respectively, while Morgan was a policy special adviser to Andrew Lansley when Lansley was health secretary between July 2010 and September 2012.
Birtwistle said the agency was founded on a new model designed to focus on achieving cut-through, rather than being built solely on the back of contacts.
"The traditional health agency model is either dead or dying," he said. "Making contact with decision makers is now really easy, but achieving cut-through is more challenging."
He added that the NHS was at a turning point and people were going to have to work harder and smarter in future to make it adapt and change in a way that will help their businesses.
"Bill Morgan has been closely involved in the reforms of health and social care and so he has a good understanding of how the system should work, and the faultlines in the new system," he added.
Incisive Health has already secured work from pharmaceutical companies, charities and health professional bodies, according to Birtwistle.
Birtwistle and Winstone were replaced as MHP's health policy team head and deputy head by joint-heads Rupert Gowrley and Rachel Rowson.