Inter's 2-1 loss to Barcelona means Lucien Favre's side finished three points above the Italian side in Group F and will be in the hat for the knockout draw, which takes place in Nyon on Tuesday (AEDT).
The impressive Jadon Sancho opened the scoring in the 10th minute before Tomas Soucek scored a deserved equaliser for the Czech side shortly before half-time.
However, Julian Brandt sealed the win with a powerful strike shortly after the hour mark, although Dortmund had Roman Burki to thank for their progression after a string of stunning saves either side of Julian Weigl's dismissal for a second yellow card with 13 minutes remaining.
Sancho and Mats Hummels went close inside the opening five minutes for the hosts, while at the other end Burki did well to keep out Peter Olayinka's close-range header.
Winger Sancho was not to be denied in the 10th minute, the England international slotting home from 10 yards after being teed up by Marco Reus, who saw a low effort repelled by the outstretched foot of Ondrej Kolar soon after.
Kolar again denied a clean-through Reus soon after before Burki superbly pushed over Lukas Masopust's deflected effort.
There was little the Dortmund goalkeeper could do to stop Soucek equalising two minutes before the interval, the midfielder sweeping home after Masopust's lofted ball had ripped open the home defence.
Brandt fizzed narrowly over and Raphael Guerreiro forced Kolar into a low save as Dortmund started the second half on top.
Sancho turned provider as Dortmund regained the lead in the 61st minute, Brandt latching onto his precise pass and lashing past Kolar from a tight angle.
Slavia continued to press, though, with Burki forced into a flying save to deny Milan Skoda while Ondrej Kudela inexplicably heading wide from point-blank range at the back post.
Weigl received his marching orders for a second bookable offence before Burki made another magnificent save to keep out Abdulla Yusuf Helal's header as Dortmund advanced.