Leading spinner Swann is being rested to allow him an opportunity to overcome discomfort in his right elbow ahead of the highly-anticipated Test series against South Africa.
Warwickshire paceman Woakes and off-spinner Tredwell have been on the fringes of the England set-up for a prolonged period of time and national selector Geoff Miller said it was always part of the plan to make the changes.
"We had earmarked the fourth and fifth matches in this series as an opportunity to rest Graeme regardless of the result at Edgbaston as we have to consider our preparations for the Test series later this month,” Miller said.
"Graeme has some pain in his elbow at the moment and we feel that an extra week’s rest will allow his elbow to settle."
He added: "Chris Woakes replaces Jade Dernbach in the squad, who has been ruled out with a side strain.
"We have played some very good cricket in the last week to lead this NatWest Series 2-0 and will need to continue that form if we are to win our seventh straight series on home soil."
Samit Patel has been released to play for Nottinghamshire Outlaws in the Friends Life t20 fixture against Yorkshire Carnegie on Friday, while Jonny Bairstow will be available for Yorkshire’s match against Lancashire Lightning at Old Trafford.
Both players will return to the squad ahead of the fourth ODI at Chester-le-Street on Saturday, with the final match of the series at Old Trafford on July 10.
England lead the five-match series 2-0 after victories at Lord’s and The Oval with the third ODI at Edgbaston having been washed out on Wednesday.
England squad: Alastair Cook (captain), James Anderson, Jonny Bairstow, Ian Bell, Ravi Bopara, Tim Bresnan, Stuart Broad, Steven Finn, Craig Kieswetter, Eoin Morgan, Samit Patel, James Tredwell, Jonathan Trott, Chris Woakes