Dortmund was beaten 3-1 on Wednesday to lose 6-3 on aggregate.
An early goal would have been the one thing Thomas Tuchel wanted to avoid, but the breathtakingly talented Kylian Mbappe put the home side in front after just two minutes.
Monaco will discover their semi-final opponents when the draw is made on Friday, with Real Madrid, Atletico Madrid and Juventus also progressing.
"We've got this out of our system now, and we're more stable emotionally", said Tuchel, who was speaking through a translator at the pre-match news conference.
Bartra was there to support his teammates in Monaco, but it was another hard night as their bus was delayed in leaving for the ground, held up by police for security reasons.
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The incident occurred a week after the first leg was postponed by 24 hours after the Borussia team bus was attacked with explosives on its way to the stadium.
"We are conscious of what occurred in Dortmund and we've it in mind". The rearranged quarter-final first leg was played just a day removed from an attack on Thomas Tuchel's team looked out of touch against the slick Monaco attack. "We stood there for 16, 17 minutes".
Before kickoff, Tuchel had told Sky: "We didn't know what was going on".
After losing 3-2 at home in the first leg, the visitors had little choice but to go forward in numbers, leaving themselves vulnerable on the counter-attack.
"In the first leg, it was impossible for us to play the game and take it seriously in the way we needed to do".
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In a match overshadowed by the Dortmund bus attack where defender Marc Bartra was injured. We'll try to be flat out in the Champions League and the league, but it's normal that we're more exhausted than teams who play less.
Monaco will meet Borussia Dortmund on Wednesday April 19.
"In the first leg it was impossible to be at a normal level, in terms of concentration".
Dembele and Reus continued to threaten, while Monaco midfielder Tiemoue Bakayoko headed over and Mbappe failed to control a good pass as the chances came thick and fast on a chilly night.
Mendy and Mbappe then combined for Lemar to add his fourth assist of the knockout stages when he set up Falcao for the former Manchester United and Chelsea striker's 25th goal of the season.
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