The role of chairman will see Lewis take responsibility for party administration, oversee the Conservative Campaign Headquarters and join the Cabinet as a minister without portfolio.
His replacement was announced on Twitter, only for the tweet to be nearly immediately deleted.
The job Ms Greening turned down, Work and Pensions Secretary, was given instead to Esther McVey, who triggered controversy when she was a junior minister in the department under David Cameron.
The new year reshuffle has been partly prompted by the forced resignation last year of May's long-time ally Damian Green, but is also an opportunity for the Prime Minister to oust people who have been disloyal or broken with her vision for government. Sir Patrick had only been chairman since 2016 but was widely blamed for last year's election result, where the Tories lost their majority.
He was also blamed for the disastrous party conference, where magnetic letters fell off during May's speech which was being televised live.
While viewed as a chance at a fresh start, the reshuffle brings risks of upsetting the delicate balance of eurosceptic and pro-European ministers. "It also allows a new generation of gifted ministers to step up and make life better for people across the whole United Kingdom", the prime minister said.
Several new faces are to be appointed to the ministerial lineup, but Philip Hammond, Boris Johnson, Amber Rudd and David Davis are among key figures likely to stay in place.
Negotiations on a transition deal to ease the break begin this month, while talks on a post-Brexit trade agreement between Britain and the European Union are set to start in March. However this will only happen if Ministers focus more on policy and delivery, and less so on personalities and rivalries.
Update: Northern Ireland secretary James Brokenshire has just resigned, citing ill-health, opening up another role for May to fill. He has failed to bring together feuding political parties in the British province, where the devolved government collapsed a year ago nearly to the day.
"(Lewis is) the type who might have the guts and inclination to take on the party internally and to be a loud media voice to take away the heat from Theresa May", a Conservative member said.
May has said she intends to stay in office "as long as people want me to serve", but past year saw numerous reports of plots to oust her - and many ministers will have their eye on a future leadership challenge.