Alexander Frei put the visitors ahead before Marco Borriello equalised only for Ghana World Cup player Samuel Inkoom to fire the Swiss champions back infront on half-time.
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Substitute Cabral sauntered through a dejected Roma defence in injury time for the third after Basel had barely mounted an attack in the second period under continued home pressure.
"The team did exactly what we planned and I'm naturally very happy," Basel coach Thorsten Fink told reporters.
"It's a really great feeling to win away in the Champions League at a top Italian team. Only an organised team has luck."
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Basel players bowed to their small contingent of fans at the end as the rest of the only half-full stadium quickly emptied with Romans shaking their heads about the side's poor season.
They can at least take comfort by glancing at the standings where Roma, Basel and CFR Cluj all have three points from three games after the Romanians lost 3-2 at leaders Bayern Munich.
BAD FORTUNE
Basel host Roma in the next match on November 3 which could be crucial to who qualifies for the knockout stages, with the Italian side's coach Claudio Ranieri hoping his luck turns.
"The team produced a lot in the second half but it's just not going our way at the moment and we've just got to keep fighting," the coach, without Daniele De Rossi and Adriano among a host of other injuries, told Sky TV.
"First half we were too stretched and not aggressive."
He had a case about the bad fortune as Roma had good pressure after the break with Francesco Totti, off colour recently as niggles dog him, back on form and going close three times.
The striker then set up Rodrigo Taddei, who rapped the post in a frantic start to the second half but the chances came and went before Cabral slalomed through the defence to seal matters.
The game had also started well for Basel with Switzerland's most famous player Frei opening the scoring on 12 minutes by thumping home from the edge of the area after Marco Streller intelligently headed the ball back.
Roma responded and Borriello, who had earlier missed a good chance, equalised with a cool clipped finish after a superb first-time ball from strike partner Totti.
Inkoom struck the killer blow just before the break when he picked up Benjamin Huggel's pass too easily in the box and rifled an angled shot into the corner.
Indebted Roma, up for sale with reports saying up to 12 parties are interested from across the globe, finished second in Serie A last term and almost snatched the title off Inter Milan but this season they lack belief and lie 13th domestically.
