Chavhanga's two tries both came in the first half as Steffan Jones converted both and added two penalties, leaving the Dragons 23-12 in front at half-time.
Jones added another while Dragons replacement Lewis Robling grabbed two penalties after the break. Harry Leonard notched his fifth penalty of the night for the hosts but there was no way back for Edinburgh.
And Bradley believes his side got exactly what they deserved on the night.
"We made too many handling errors which made it too easy for the Dragons," said Bradley. "We didn't put enough pressure on them."
Bradley's side now sit 11th in the RaboDirect PRO12 table, just four points above bottom side Aironi with four matches remaining.
Edinburgh now face the Scarlets at Murrayfield on Friday before welcoming Toulouse to the Scottish capital for their Heineken Cup quarter-final - only the second time they have reached that stage of the competition.
Bradley added: "We now face a huge challenge with the arrival of a high-flying Scarlets side, as we seek to produce the kind of performance our fans are familiar with and can allow us to gain momentum going into the historic quarter-final with Toulouse."
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