The win gave Jorge Fossati's side an 8-2 aggregate victory and booked a two-legged final with Fluminense - a repeat of the 2008 Copa Libertadores final, won by the Ecuadorians at the Maracana. Bieler netted a penalty in the 18th minute to level the tie after Liga's 2-1 first-leg loss in Montevideo, following a foul by Diego Sosa on Neicer Reasco.
Nine minutes later Carlos Espinola headed home an Edison Mendez corner to double the lead. Seconds before half-time, Miller Bolanos made it 3-0 before Mendez widened the gap further after William Araujo's long-range effort set him up.
In the 77th minute, Ulises de la Cruz came off the bench to secure Liga's triumph with another goal from outside the box and in the final eight minutes, Franklin Salas twice set up Bieler. Mendez told Fox Sports: "We never got desperate. Claudio's first goal helped us play a great game because it came quite quickly.
"We scored a lot of goals, but we also played a great game tonight. To reach the final of a continental championship is the most difficult thing. Then we'll try to plan how to win them."
Bieler said: "We never give up. That's the most important quality of the team. I hope we can beat Fluminense again and bring joy to our fans."
The first leg of the Copa Sudamericana final will be played in Quito next Wednesday.