Jurgen Klopp's men dominated proceedings from start to finish at the Commerzbank-Arena, but had to wait until the 83rd minute to claim victory through Aubameyang's 14th goal of the season.
The outstanding Henrikh Mkhitaryan sent in a corner that was flicked on at the near post and the Gabon international nodded home from close range to move Dortmund one step closer to the final in Berlin.
Injury-hit Dortmund were without a host of first-team regulars - including Marco Reus and Sven Bender, who picked up thigh problems in the 5-1 hammering of Werder Bremen on Saturday - but they still controlled the opening exchanges.
Mkhitaryan, who scored twice in the visitors' 2-1 Bundesliga win at the Commerzbank-Arena in September, had their first attempt on goal after 12 minutes when he fired wide from 18 yards.
Dortmund continued to probe, with Mkhitaryan a constant menace, and Kevin Grosskreutz forced Kevin Trapp into a fine save with a shot from outside the area just after the half-hour mark.
Aubameyang was next to go close, as his 30-yard free-kick drifted wide of Trapp's right-hand post.
Grosskreutz again spurned an opportunity to make Dortmund's dominance pay six minutes before the break as he could only shoot over the bar after being played in by Nuri Sahin.
Frankfurt were clinging on for the half-time whistle, and rode their luck once again when Robert Lewandowski opted to pass instead of swivelling and shooting from close range after receiving the ball with his back to goal on the stroke of the interval.
Armin Veh's side regrouped at the break and came out with far more attacking intent, but it was Dortmund who again had the first meaningful effort on goal.
Sahin swung in a pinpoint free-kick delivery 11 minutes after the restart, but defender Sokratis Papastathopoulos could only divert his attempt over the bar after sliding in at the back post.
Dortmund's supremacy finally told seven minutes from time, though, as they made the crucial breakthrough courtesy of Aubameyang's close-range header.
Klopp's side, who last won the Pokal in 2012, then comfortably held on to their slender lead to seal their semi-final berth.
The sides meet again in the Bundesliga on Saturday and, on this evidence, Frankfurt will be in for a long afternoon at Signal Iduna Park.
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