Reporting directly to CEO George Black, Edgar oversees a team of 22 at the authority.
Edgar - the naval ships comms manager at BAE Systems - replaces ex-head of corporate comms Caryl Jackson, who left three months ago to take a career break. Since her departure, responsibility for internal comms has moved to the e-government team.
The post was previously held by John Brown, brother of the Chancellor, who left in 2004.
‘I want the rest of the world to know that this is a great place to live, visit and do business,' Edgar said.
Edgar left BAE Systems this week after a year in the role. He was head of comms at the Scottish Labour Party for more than four years and was previously the producer of BBC Radio Scotland's Politics Tonight.
Glasgow is the third largest city in the UK. Issues include the planned sell-off of £200m worth of land and buildings to fund spending on services.
Other major projects include regeneration of the Clyde waterfront.
The city is bidding to host a super-casino and the 2014 Commonwealth Games.