"It has been an absolute privilege to have been in charge ... these last 30 months and we go to France tomorrow in good spirits with a strong desire to do well in the next few weeks."
O'Neill took over in November 2013 and guided Ireland to the European Championship via a playoff victory over Bosnia.
The new deal means O'Neill will be handed the task of helping Ireland end what will be a 16-year absence from the World Cup, with their last appearance in the tournament coming in 2002.
Delaney said the new contracts marked a "great day" for Irish football.
"The board of the FAI were unanimous in offering Martin and his full management team a renewal," he added.
"Since Martin has taken over we have achieved the objective of qualifying for the Euros and he has led the project in helping to develop some of the emerging talents in Irish football."
Fellow members of O'Neill's management team, Steve Guppy, Seamus McDonagh and Steve Walford, have also agreed new deals.
Ireland are in Group E at the European Championship where they will face Belgium, Italy and Sweden.
(Reporting by Toby Davis; Editing by Tony Jimenez)
