Jones, 27, joined Edinburgh from Newcastle Falcons in the summer and impressed during the warm-up matches against Watsonians and Harlequins.
He subsequently made his first start against Ospreys in October and featured in the Heineken Cup win over Ulster in October; overall recording five appearances.
But his departure has now been confirmed ahead of the English transfer deadline and Moffat paid tribute to the player.
Moffat said: "We are sorry to see Steve go - he is a quality rugby player and a good character to have in the squad.
"We wish him well in his future rugby career and hope that he enjoys success with Exeter."
Jones admits he enjoyed his time at Edinburgh and has now set his sights on helping Exeter reach the English Premiership.
Jones said: "I've enjoyed my time at Edinburgh and the city itself is stunning. I've enjoyed living there for a short period of time and I've made a lot of friends.
"But I've decided to move on. My main goal now is to help Exeter to get promotion into the Premiership. I'm really looking forward to the challenge."
The Chiefs are in the same British and Irish Cup Pool as Gael Force, the team consisting of professionals from Edinburgh, Glasgow and members of the National Academy.
As a result, Jones says it would have been strange if he had featured for the Force against Exeter when the two sides met only a couple of months ago.
He said: "That would have been quite bizarre if I had played against them for Gael Force a few months ago, but now I'm looking forward to playing for Exeter in the British and Irish Cup."
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