Alexander Frei stunned the hosts with a rasping strike to put the Swiss side in front in the 12th minute before an exquisite finish from Marco Borriello levelled the scores up nine minutes later.
But an emphatic effort from defender Samuel Inkoom restored the hosts' lead, and Claudio Ranieri's side were impotent in a second half which ended fittingly with a breathtaking individual effort from substitute Cabral to seal an inspired victory for the Swiss.
Roma have only kept two clean sheets in their last 19 Champions League matches, and the hosts were punished for their lethargic opening as Frei provided a touch of class in the final third.
The Swiss striker latched on to Marco Streller's knockdown before picking out the far corner of the net with a crisp, unerring effort which left Bogdan Lobont comprehensively beaten.
Ranieri's side gradually established themselves in the first half, and it took a sublime piece of skill from Borriello to cancel their arrears: the forward raced through on goal from Simone Perrotta's incisive through ball, before lifting a deft finish over the diving Franco Costanzo.
The possession was shared thereafter as both sides hustled and harried to good effect in a congested midfield, but Basel were back in front on the stroke of half-time to further frustrate the hosts. Inkoom careered forward from his right-back berth before cutting inside and hammering a piledriver into the top corner of Lobont's net with aplomb to restore his side's lead.
Francesco Totti was an ineffectual presence in the first half, but came out fired up in the second and proceeded to be at the forefront of everything going forward for the hosts, who returned after the break with a new sense of urgency.
Marco Cassetti surged forward down the right, before Taddei seized the initiative and drilled a low shot which left Costanzo flat-footed, only to see his effort cannon off the far post with Perrotta unable to convert the follow up.
Former Arsenal forward Julio Baptista was introduced in an attempt by Ranieri to add a hefty physical presence to the Roma forward line, and the Brazilian almost made the required impact. Baptista picked up the ball out wide on the left and whipped over a pinpoint delivery which was met by an unmarked Taddei just 10 yards from goal, but the midfielder could not hit the target and squandered the opportunity.
Another close-range headed opportunity was wasted by defender Philippe Mexes six minutes from time as the Frenchman climbed above Beg Ferati but his attempted finish was poor.
The final effort from the increasingly dejected hosts came from Totti, whose thunderous drive from a 25-yard free-kick was superbly parried behind for a corner by the excellent Costanzo.
A memorable victory for Fink's side was wrapped up in stunning fashion in stoppage time as substitute Cabral weaved mesmerically through four challenges before coolly slotting the ball beyond Lobont to secure the three points.
Basel's improbable win lifts the Swiss side above Roma in Group E, with the Italian side rock bottom while Bayern Munich maintained their unblemished record with a third successive victory.
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