Following a 10-minute delay to kickoff due to the Dortmund team bus being held up, Raheem Sterling and Kai Havertz struck either side of the break to erase BVB's 1-0 lead from the first leg and ease some of the pressure on Potter's shoulders.
Already being forced to cope without first leg match-winner Karim Adeyemi while number one goalkeeper Gregor Kobel warmed the bench, Dortmund's injury list welcomed its newest member just five minutes in, as Julian Brandt pulled up with a hamstring problem and had to be replaced by Giovanni Reyna.
Despite that early setback, Dortmund soon began to dictate the tempo as the first half progressed, dominating possession and testing Kepa Arrizabalaga with a well-struck free kick from Marco Reus in the 17th minute, which the Spaniard got a strong hand to.
Chances soon began to fall the hosts' way, but in a tale as old as time, their luck was just not in, as Havertz's half-volley in the 28th minute struck the inside of the post before trickling across the line and somehow staying out.
The woodwork would then come to the aid of the German in the 38th minute as he fired home what Blues fans briefly believed to have been the opening goal, but the linesman's flag belatedly went up against Sterling from earlier in the move.
Joao Felix also saw a close-range effort kept out by Alexander Meyer in the 40th minute in a familiar sight for the restless Blues faithful, whose groans were audible when Sterling miskicked horribly from Ben Chilwell's cross three minutes later.
However, that unintentional mistake handed Sterling a second bite at the cherry, and the Englishman composed himself to fire home into the roof of the net at the second attempt to level the tie on aggregate.
The architect of the first goal, Chilwell - who was thoroughly enjoying himself on the Stamford Bridge turf - won his side a penalty four minutes into the second half when his cross hit the outstretched hand of Marius Wolf, and a spot kick was awarded following an on-field review.
Havertz's profligate streak was seemingly set to continue as he struck the post before Salih Ozcan cleared, but the Turkey international was one of a few Dortmund players to have encroached, and a retake was subsequently awarded.
Taking an identical stuttering run-up, Havertz went for the same corner and made no mistake on his second attempt, putting Chelsea 2-1 up on aggregate before the Dortmund onslaught began.
Jude Bellingham spurned a magnificent chance to claw Dortmund level on aggregate in the 58th minute - sidefooting wide from close range following a lucky break of the ball - but it was Chelsea who had the ball in the back of the net in the 76th minute through Conor Gallagher, although Sterling had gone too early again.
Chelsea comfortably survived six minutes of injury time to seal their spot in the last eight, and next up for a revitalised Blues side is a trip to Leicester City in the Premier League on Saturday afternoon.
Meanwhile, Dortmund resume their Bundesliga title fight away to Schalke on Saturday evening.