In front of a small but vocal support at Stade Louis II, the tie's standout striker, Kylian Mbappe, scored the opener inside three minutes, following up on the rebound after Benjamin Mendy's stinging drive was too hot for Roman Bürki to hold.
Dortmund's No.7 won a team-high five take-ons and created two chances, including his assist for Marco Reus' goal early in the second half - a powerful run from deep and ideal square which sealed his sixth Champions League assist of the campaign - behind only Neymar (eight). "We were always looking for the extra goal, that is the way we play, it is in our DNA".
Reus, who missed the first leg through injury, is expected to be made available for the second leg against Monaco and could be a major factor in their bid to do the impossible.
Monaco last reached the semifinals in 2004, losing the final to Porto.
Monaco joined Real Madrid, Atletico Madrid and Juventus in the last four of Europe's premier cub competition.
Borussia Dortmund head to Monaco with the task of overturning a 3-2 first leg deficit it what is set to be a hard night for the Germans. After every step we've taken we've looked at it and we've kept working.
The initial leg was postponed after an assault on Dortmund's team trainer but Jardim considers the Bundesliga team could have recovered their focus for his or her visit to Monaco. Unfortunately for them, Monaco have simply not allowed it.
"I saw the match on Saturday [a 3-1 triumph against Eintracht Frankfurt], they played nicely. They recover well. It's normal we're more exhausted than those who play less".
"We know that tomorrow is a final, it will be necessary to respect our principles of play. We will give everything in both", added the coach.
Borussia Dortmund face an uphill task to overturn a 3-2 deficit against French rivals Monaco and keep their minds on football after the first leg of their Champions League quarter-final was marred by an attack on the German team's bus.
After the bad events in Dortmund last week, the German side's players again endured security concerns, as they were forced to sit on the team bus for an extra 20 minutes while it was inspected.
Tuchel said the German club's bus was scheduled to leave one-and-a-half hours before kick-off and "everyone was there ready to go". Kickoff was briefly delayed.
Apart from the long-term injured Mario Goetze, Andre Schuerrle and Sebastian Rode as well as Marc Bartra, who was wounded in the bomb attack on the team bus manager Thomas Tuchel has a full contingent to choose from. "We're the only ones who can understand what happened since we were there".
Borussia Dortmund have lost eight of their last 11 games in the Champions League knockout stages (W3). Attacking midfielder Marco Reus pulled one back in the 48th minute.