Greig Tonks kicked a couple of early penalties for the away side, but Fenby caught an interception and ran in to get London Irish their first points after 11 minutes.
Irish then found a gap straight from the kickoff and Fenby grabbed his second just one minute later, Shane Geraghty adding the conversion this time.
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It was all Edinburgh after the break but they failed to make their dominance pay, and Aseli Tikoirotuma crossed for Irish halfway through the half.
Fenby then finished off his hat-trick with five minutes to go, following good work from Lewington which allowed the winger to cross untouched.
The wins sees Irish move to the top of their pool for the moment, with Edinburgh dropping down to third.