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The visitors looked comfortable early on, scoring three tries inside the opening 25 minutes, but efforts from Chris Pennell and David Lemi got the Warriors back into contention shortly before the break.
Jack Yeandle extended Exeter's lead with a try either side of half time, seemingly putting the win beyond doubt.
However, the hosts fought back as Lemi claimed his second while Agustin Creevy also forced his way over, but they came up just short of Exeter's total.
The result moves the Chiefs up to seventh, while Worcester remain bottom.