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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"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."
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.
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."
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.
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.
