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Roma had started the match well, with Gervinho seeing an early shot saved, but the Serie A outfit soon found themselves two goals down inside the opening quarter of an hour.
BATE went ahead when Ihar Stasevich was able to head home from close range after a Mikhail Gordeichuk strike had hit the crossbar, before three minutes later, Mladenovic unleashed an effort from distance that deceived Wojciech Szczesny and found the back of the net.
Roma responded through Radja Nainggolan, who had a 25-yard strike saved by the goalkeeper, but shortly after the half-hour mark, BATE found a third.
Questions will again be asked of Szczesny, who failed to deal with another shot from Mladenovic, who registered his second in the space of 20 minutes.
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Roma needed a reply soon after the restart, but Alessandro Florenzi was denied when his goal-bound effort was blocked, while they were fortunate to avoid falling further behind when Evgeni Yablonski failed to capitalise on a clear opening inside the penalty area.
Midway through the second half, Roma pulled a goal back thanks to Gervinho, who provided the cool finish after being being played in by Iago Falque.
With 15 minutes left, Mohamed Salah missed a chance to bring Roma back to within one as he skied his shot into the stands, but on 82 minutes, Vasilis Torosidis netted the Italian side's second after converting from close range.
Roma threw men forward in search of an equaliser but the Belarusian club held on to to not only dent Roma's hopes of qualification, but boost their own as they target second place behind likely group winners Barcelona.