Emre Can struck a stunning goal from the halfway line and skipper Marco Reus clinched the late winner for the five-time German Cup champions.
Despite Bochum's impressive form at home, having won five consecutive matches since Thomas Letsch took over as manager in September, Dortmund emerged victorious in a intense and demanding match.
Can's quick thinking just before halftime gave Dortmund the lead. Bochum's goalkeeper, Manuel Riemann, tried to prevent a Dortmund attack by rushing outside his box, but his clearance landed in Can's possession.
The former Liverpool and Juventus midfielder controlled the ball with one touch and then sent a long-range effort into the net, leaving Riemann helpless.
Can later remarked to Sky Germany, "I actually didn't hit it that well, but in the end, it was in. It was a beautiful goal."
English teenager Jamie Bynoe-Gittens, starting for the first time for Dortmund, was nearly able to extend their lead when he leapt to meet a cross in the 59th minute, but the ball made contact with his elbow.
After a lengthy VAR review, Bochum was awarded a penalty, and Austrian midfielder Kevin Stoeger equalized with a shot down the middle.
Dortmund manager Edin Terzic brought on Reus three minutes later, and the two-time German Cup winner scored three minutes after that, converting a perfect pass from Jude Bellingham into an empty net.
Terzic, who won the Cup as Dortmund's interim manager in 2020-21, praised his team for winning a "huge Cup fight" and credited Bochum for making their victory a challenging one.
Letsch, who experienced his first home defeat as Bochum manager, told German TV, "we have to put that one behind us now, but we can leave the pitch with our heads held high."
Dortmund will join the likes of Bayern Munich, RB Leipzig, Stuttgart, Union Berlin, Freiburg, Eintracht Frankfurt, and second-division Nuremberg in the quarterfinals.