Ciro Immobile, under-fire for his lack of goals this season, tapped the visitors in front after 33 minutes before Marco Reus rifled in the second goal after a cut-back by the Italian a minute before the break.
Apart from two early chances for the hosts, the second half was a subdued affair with Dortmund focused on holding on to their lead rather than adding to their goal tally.
They made it 3-0 when a bad clearance by keeper Philipp Tschauner fell straight to Shinji Kagawa and the Japanese made no mistake as he gave his side a boost of self belief ahead of Saturday's league game with Bayern Munich.
"It was important to advance after the few weeks we have had," Dortmund captain Mats Hummels told ARD television. "It is a matter of confidence for us but we can get it back."
Dortmund, last season's Bundesliga and German Cup runners-up, have suffered six defeats in nine games to slide to fourth from bottom in the table.
Werder Bremen coach Viktor Skripnik made a winning start as his new club eased past third tier Chemnitz 2-0.
Ukrainian Skripnik replaced the sacked Robin Dutt on Saturday, with the former champions anchored in last place in the league.
Fin Bartels put them ahead in the first half and Franco Di Santo drilled in the second goal after a fine through ball from Clemens Fritz after the interval.
Hanover were handed a surprise 2-0 defeat at relegation-threatened second division club Aalen and Hertha Berlin fared even worse, crashing out to third tier Arminia Bielefeld 4-2 on penalties after a goalless 120 minutes.
Holders Bayern are in action on Wednesday against Hamburg SV.
(Reporting by Karolos Grohmann, editing by Tony Jimenez)
