The tie was delicately poised from the first leg, with a Luuk de Jong penalty cancelling out an opener from Donyell Malen, leaving the clash level at 1-1 heading into Wednesday's fixture.
Jadon Sancho and Marco Reus netted to secure a two-goal success in Germany, with BVB clinching their position in the final eight of the competition courtesy of a 3-1 aggregate win.
Dortmund threatened to take the lead in the second minute of the contest through Ian Maatsen, but the 22-year-old's effort was kept out by PSV goalkeeper Walter Benitez.
BVB were not to be denied, though, with on-loan Manchester United forward Sancho making the breakthrough less than a minute later.
PSV failed to clear their lines, and a set-up from Julian Brandt allowed Sancho to drive the ball into the bottom corner of the net for his second goal since returning to the club during the January window.
The goal was his first in the Champions League in 840 days, and it allowed the Bundesliga outfit to take a 2-1 lead in the tie, with PSV not given time to settle into the second leg in front of a passionate home support.
Niclas Fullkrug and Malen also had efforts in the period that followed, with Dortmund looking for a second on the night, with the hosts playing at a furious tempo.
Malen came close to doubling Dortmund's advantage in the 27th minute when he worked a shooting position before driving the ball towards the bottom corner, but it was just narrowly wide of the post.
Dortmund goalkeeper Gregor Kobel made a smart save in the 31st minute to keep out a drive from Johan Bakayoko, with his effort taking a slight deflection off Marcel Sabitzer en route to goal.
PSV were pushing for a leveller in the latter stages of the first half, with Bakayoko coming close again before Kobel denied Guus Til, as Dortmund were almost made to pay for some lacklustre defending.
Malen had another half-chance in the 40th minute after finding space in a dangerous area, but his effort hit the side-netting, before De Jong missed the target down the other end in the 43rd minute, and Sancho's goal proved to be the difference between the two sides at the break.
The first opportunity of the second period also fell to Malen, with his shot taking a deflection off Sergino Dest before being kept out by Benitez.
PSV came so close to levelling in the 53rd minute when Hirving Lozano lined one up from distance before unleashing a fierce strike, and his effort came off the outside of the post.
The visitors were the better side as the second period continued to progress, with Dortmund boss Edin Terzic cutting a frustrated figure on the touchline, and the hosts were anything but comfortable in the tie.
Fullkrug had a half-chance to net a second for the hosts in the 64th minute after meeting cross from Malen, but Benitez was on hand to make a smart save, before Kobel kept out a powerful strike from Bakayoko down the other end of the field.
Dortmund thought that they had scored a second in the 77th minute through Fullkrug, but the striker's effort from a Reus free kick was ruled offside following a VAR review.
PSV were therefore given a boost entering the final stages of normal time, with the Dutch outfit needing a goal to force an additional 30 minutes, but Jordan Teze and De Jong both missed clear openings late on, before Reus secured BVB's spot in the final eight by sliding the ball under Benitez after a mistake from Isaac Babadi.
Dortmund's attention will now switch back to the Bundesliga, preparing to host Eintracht Frankfurt on Sunday afternoon, while PSV will continue their Eredivisie campaign against FC Twente later in the day.