Darren Edwards’ men trailed 14-9 as full time approached at the Kassam Stadium.
But they stormed to victory with 33 unanswered points in the final 10 minutes of the encounter.
Ryan Davis and Steffan Jones exchanged penalties in a relatively eventless first half as Welsh edged to a 9-6 lead.
Jones hit-back following the break but Welsh went back in from after Ed Williamson was driven over from close range.
Gavin Henson, returning from injury, missed the conversion and the Exiles soon capitulated as the Dragons ran riot.
Jones slotted a quick-fire hat-trick of penalties as the Welsh region surged ahead.
Hallam Amos and Jack Dixon led the charge with ball in hand and Amos danced over for a crucial try.
The Dragons continued to dominate the closing stages and Ieuan Jones bulldozed over following a sustained period of pressure.
And the result was put beyond doubt when Jevon Groves intercepted a stray pass to race over.
Jones added the extras and slotted his seventh penalty with the last play – to take his haul to a Dragons record-breaking 27-points and level Jason Tovey’s record of seven penalties in a game for the region.
