After Fabinho missed an early penalty for Leonardo Jardim's side, Mbappe made the breakthrough on 19 minutes and Sven Bender's own-goal 16 minutes later put them in command.
But Dortmund improved vastly in the second half and hit back through Ousmane Dembele, although Mbappe secured the win with his second goal on 79 minutes before Shinji Kagawa pulled another one back late on to keep the German team in the tie.
The previous evening's events clearly affected Dortmund as they struggled to find any sort of rhythm in the rescheduled fixture.
The first chance came the way of Thomas Tuchel's men in the 11th minute, though, when Lukasz Piszczek sent Pierre-Emerick Aubameyang through on goal with a fine pass, but the striker fired high and wide of the target from an angle on the right.
But the Ligue 1 leaders were causing problems with their pace and crisp passing, and they were presented with a glorious chance to open the scoring from the penalty spot on 16 minutes after Mbappe went down under a challenge from Sokratis.
However, Dortmund fans breathed a collective sigh of relief as Fabinho steered his spot-kick wide of the left post.
Monaco quickly put that disappointment behind them to break the deadlock three minutes later. Bernardo Silva engineered a breakaway with a surging run from the right and slipped in Thomas Lemar on the left.
The French midfielder sent in a low cross towards the back post, where Mbappe converted from six yards out off his thigh.
Shinji Kagawa wasted a good opportunity to pull the hosts level on 31 minutes as he stabbed the ball wide of the left post after Matthias Ginter did well to get into the box from the right and pick him out.
The visitors made them pay just four minutes later, with Sven Bender heading Andrea Raggi's cross past his own keeper in an attempt to prevent the ball reaching Radamel Falcao.
The Principality outfit deservedly led at the break, but BVB looked much better in the second half and Dembele looked to halve the deficit from a 46th-minute free-kick from just outside the area that he curled well wide of the right post.
Dembele took a much easier chance to pull one back on 57 minutes as he tapped home Kagawa's square pass from close range after Aubameyang found the Japanese playmaker on the left with an audacious back-heel.
Suddenly Monaco were battling the keep hold of the ball as Dortmund looked refreshed after the interval, although they were unable to make the pressure count.
But a mistake from Piszczek let in Mbappe 11 minutes from time and the French teenager obliged, intercepting a poor pass to race through on goal before bending a shot into the top-right corner of the net from 18 yards out.
The Bundesliga club continued to press in the closing stages and they were rewarded on 84 minutes when Kagawa displayed good skills to control Nuri Sahin's pass before cutting back onto the left foot and slot past Danijel Subasic from seven yards out.
Aubameyang missed a late chance to restore parity as he directed a header over the crossbar from Dembele's delivery on the right in stoppage time.