The Ameobi brothers, Shola, 31, and 20-year-old Sammy, combined in breathtaking style three minutes into the second half to set up the Frenchman to secure the narrowest of victories.
It was a far from comfortable evening for a much-changed Newcastle side, although they might have been in control of the game long before Obertan struck had Papiss Cisse managed to convert any of the chances which came his way.
Alan Pardew's men were made to fight all the way as the Pro League leaders gave a good account of themselves, with Steve Harper making a fine second-half save from Lior Rafaelov to preserve his clean sheet.
But they made it to the final whistle without mishap in front of a crowd of 33,124 at St James' Park to take their Group D points tally to a satisfactory seven from three games and leave themselves on the brink of qualification for the last 32.
Pardew took the bold decision to rest no fewer than nine of the men who started Sunday's 1-1 derby draw at Sunderland while by contrast, opposite number Georges Leekens made just two changes after Saturday's first league defeat of the season at Leuven.
What looked like a tricky affair on paper proved to be just that as Brugge gave as good as they got in an open first half.
They arrived set up to hit the Magpies on the break and although dangerous striker Carlos Bacca was clearly labouring as a result of the training-ground injury he picked up earlier this week, the pacy Maxime Lestienne proved a major threat down the left.
He forced a third-minute save from Harper within three minutes of kick-off and shot across goal and narrowly wide of the far post a minute before the break as he repeatedly stretched right-back Davide Santon and central defender Steven Taylor.
But in the meantime, Newcastle had created more than enough openings of their own to have taken a lead and, with a little better fortune, a commanding one.
Cisse, who has endured an unproductive start to the season by his standards, might have struck with 14 minutes gone when he met Shane Ferguson's left-wing cross with a firm downward header, but saw keeper Bojan Jorgacevic block with his body.
Sammy Ameobi tested Jorgacevic with a rising 25-yard drive three minutes later and dragged a 31st-minute effort just wide after cutting inside from the left with the help of some neat footwork.
However, the visitors were handed a golden opportunity to make the breakthrough three minutes later when Cheick Tiote played a blind pass to nobody in particular and saw Lestienne run in on goal but fire wide, with the help of a deflection off the covering Taylor.
The home side finished the half with a flurry with Cisse, who had earlier seen appeals for a penalty turned away following a clumsy aerial challenge by Bart Buysse, at the centre of it.
The Senegal international headed over from a 39th-minute Ferguson corner and failed to make the most of a superb piece of control by missing the target three minutes later.
But it took a fine diving save by Jorgacevic to deny him in injury time after Vurnon Anita had set him up to volley towards the top corner.
Pardew replaced Anita with Shola Ameobi at half-time and he and his younger brother combined with stunning effect within three minutes as the Magpies roared into the lead.
Shola's neat flick found Sammy, and he produced an outrageous piece of skill to play the ball into space before laying into the path of Obertan, who thumped it past Jorgacevic and inside the far post.
The lead might have lasted just four minutes, however, had 37-year-old Harper not shown commendable agility to get down at his near post and turn Rafaelov's left-foot shot away for a corner.
Brugge were responding well and Harper was pleased to see a 59th-minute shot from substitute Ivan Trickovski drift well wide before fielding a toe-poked effort from Bacca seconds later.
Harper had to be alert to pluck Lestienne's 62nd-minute cross off Bacca's toe after Santon had gifted possession to Vadis Odjidja, and the Colombian sent a downward header into the ground and over the bar from another Lestienne delivery seconds later, although an offside flag spared his blushes.
Newcastle had lost all composure in the face of a sustained assault, but gradually regained it with Fabricio Coloccini and Yohan Cabaye having joined the fray to edge their way towards the final whistle and the three points.