The Premier League outfit will allow the Grecians to keep all the gate receipts from this weekend's clash whereas clubs normally share the proceeds.
Swansea avoided relegation from the Football League on the final day of the 2002/03 season at the expense of the Grecians and have gone on to become a bona-fide Premier League side and qualified for Europe after lifting the Capital One Cup last season.
"We're hugely grateful to Swansea - not only for agreeing to put on an exhibition here at St James Park, but also for allowing us to use this as a much-needed fundraiser for our football club," chief executive Julian Tagg told the club's official website.
"Thanks to Swansea's generosity, all Exeter fans that come to the game will not only be treated to a chance to see Swansea's superstars, but also to help out their own club financially in what we all recognise as challenging times.
"This shows an understanding and camaraderie between two clubs that had contrasting emotions at the end of that mutually gruelling campaign 10 years ago."