Trailing 3-2 from their home first leg, which was put back by a day following an attack on Dortmund's team bus, Thomas Tuchel's men had it all to do at Stade Louis II.
And their cause was not helped by two Monaco goals inside 17 minutes, Kylian Mbappe - who scored twice at Signal Iduna Park - pouncing on Roman Burki's poor parry and Radamel Falcao heading in a second.
Reus gave cause for optimism with a goal early after the break, but Valere Germain netted just 22 seconds after replacing Mbappe in the closing stages to seal a 6-3 aggregate win for Monaco.
Speaking to reporters after the match, Reus said: "We did not start well, we did not enter the challenges and failed to put pressure on them and it quickly was 2-0.
"Still, it wasn't over then, as we could have scored three.
"We started well in the second half but did not manage [to score] any other goals."