Having fallen behind to an early Grant Pointer try, the visitors registered 23 unanswered points, with wingers Joe Ajuwa and Nick Scott both crossing the Pirates line.
A penalty try reduced the Exiles' lead to 23-14 by the break but two tries in the space of as many minutes late on at the Mennaye Field sealed a famous victory at a venue where league leaders Bristol had seen their title challenge come to an end earlier this month.
Hudson Tonga'uiha and James Lewis were the second-half try scorers, while full back Alex Davies converted all four tries and kicked a brace of penalties, with fly-half Gordon Ross adding a drop goal.
London Welsh will now take a 16-point lead into the second leg next Wednesday, although victory won't currently be enough to see them promoted to the top division after an RFU report yesterday ruled that the club hadn't met the minimum standards criteria.
With Old Deer Park unsuitable for Aviva Premiership rugby, London Welsh were keen to play their home fixtures at Oxford's Kassam Stadium.
Question marks have been raised by the RFU regarding primacy of tenure at the football stadium but English rugby's governing body have stated that the application for promotion was turned down for 'various failures'.
Chairman Beleddyn Phillips insists that London Welsh are considering an appeal, however, if they can finish off the job on the field by getting past the Pirates again in a weeks' time.
"In the event that we do win the Championship, we will do what we can to make sure we take our place in the top rung of English rugby," said Phillips.
"We'll give it all we can both off the pitch and on the pitch.
"We have to await the detailed finding from the RFU. I'm surprised and disappointed because the Kassam Stadium is a top-notch venue. There's no questioning the quality of the stadium as otherwise we would not have signed the agreement.
"I'm at a slight quandary as to why that may not have satisfied the technical criteria, which is why I'm awaiting the details with interest."