The first round tie saw the pro Cheshire side have to work hard in the opening quarter as Fury stunned the home team and their fans by going 11-5 ahead after three minutes, before a 13-0 run for the home side saw them go 18-11 ahead.
Fury's Trey Whitley, a former Black Hills State NCAA player, then hit his second three of the game to bring it back to four points deficit with three minutes of the quarter remaining and Fury still very much in the match.
Phoenix then pulled out to a 23-14 lead, led by GB star Teddy Okereafor (14 points first quarter), but with with two minutes of the period remaining Falkirk' s Bantu Burroughs hit a long range three to bring the gap back to six.
Whitley hit his third three of the quarter as the first period closed out at 30-20 in Cheshire's favour.
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But that was as good as it got for Fury, with Cheshire dominating the second and third periods.
Fury's Whitley was the games top scorer on 22 personal with Bantu Burroughs the second highest scorer with 21 points.
Fury's Adnan Jalil and Zigy Dauksas shared 16 points with Marko Mladenovik on four and Evaldis Burksaitis on twos.
Fury head coach John Bunyan said: “Full credit to the effort and commitment our players made, in what was always going to be a very tough game.
“Chester are a great organisation with a solid standing in the British pro league. This dates back to when Mike Burton coached the side and it was special for me to speak to him, as there are a lot of similarities between our clubs, particularly in getting players on a pathway from youth to pro ball.
"I also think it’s a very important part the British basketball structure that opportunities are afforded to the very best clubs in Scotland, England and Wales, who are not in the BBL, to take part in a National club event, such as the BBL Trophy.”