The Lithuanian team improved to 3-2 and at least temporarily climbed above Proteas EKA AEL, who were to take on pool leaders EiffelTowers Den Bosch later Tuesday.
Austrians Swans dropped to 0-5 with the setback.
The game was tied at 11-11 in the first quarter when Siauliai's Virginijus Praskevicius passed to Stefhon Hannah who scored on an alley-oop.
That basket started a 10-1 run that put the visitors in front to stay.
Included in the spurt was a three-point play by Vitalijus Stanevicius.
Allianz did a good job of not allowing the visitors to run away for a convincing victory, though.
Elijah Palmer's three-pointer cut the Siauliai advantage to 59-53 heading to the final quarter.
Robertas Giedraitis' men did a good job of staying on top, however, and ultimately claimed a 10-point victory.
Siauliai's Praskevicius led all scorers with 19 points.
Saulius Kuzminskas, player of Siauliai:
"This game was very important for us if we want to qualify for the next round. So it was a good win. We gave our best and we are happy right now. We hope we can win our next game as well."
Bob Gonnen, coach of Allianz Swans:
"We missed half of our free throws and they had 12 rebounds more than we did, that was the game there."
Peter Hütter, player of Allianz Swans:
"We lost the game at the free throw line, period."