BATE went 3-0 up in the first 30 minutes through Ihar Stasevich and two from Filip Mladenovic.
The Serie A club hit back after the break and reduced the deficit with strikes from Gervinho, after 66 minutes, and Vasilis Torisidis eight minutes from time.
Roma's Alessandro Florenzi also hit the crossbar late on but BATE held out to move on to three points in Group E. Roma are bottom of the section with one.
BATE were almost written off when they lost their first game 4-1 to Bayer Leverkusen.
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And maybe even they didn't expect to be three goals to the good by half-time here.
Stasevich, the midfielder and captain, opened the scoring after eight minutes with a close-range header after an effort had come back off the bar.
Roma keeper Wojciech Szczesny was caught frozen four minutes
later when left-back Mladenovic fired a surprise long-range shot low into the net, with referee Mark Clattenburg rebuking Lucas
Digne for wearing a ring on his finger just moments before.
Mladenovic sent the home crowd into raptures for the third, when he again broke down the left, cut into the area to collect a pass from Maksim Volodko, and sent the ball just under the crossbar with a powerful shot.
The Giallorossi, missing Francesco Totti as well as the influential Seydou Keita and Edin Dzeko due to injuries, pulled one back as Gervinho slotted home following a slick combination between Mohamed Salah and Iago Falqué.
Greek defender Torosidis then smashed in from close range and Florenzi almost helped complete an incredible comeback by clattering the woodwork with a curled shot from the edge of the area.
(Editing by Stephen Wood)
