Up next for Dortmund is the first leg of their round of 32 Europa League tie against Atalanta on Thursday.
Runaway German Bundesliga leaders Bayern Munich continued their relentless progress towards a sixth successive title with a 2-1 win over Schalke 04 on Saturday, extending their unbeaten run against the Gelsenkirchen side to 15 meetings.
The hosts got off to a flying start when Robert Lewandowski scored for the 11th successive home game, in the sixth minute.
The striker moved to the Bundesliga club on transfer deadline day from Chelsea on a loan deal for the rest of the season.
Franco Di Santo stunned the Allianz Arena by equalising in the 29th minute, burying his shot after Leon Goretzka mis-kicked from Breel Embolo's cross.
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A tiresome match was lit up by Dortmund's second in the dying stages when Germany's World Cup victor Mario Goetze netted after coming off the bench.
While everyone at Borussia Dortmund celebrated a fourth German Cup trophy in May 2017 Marco Reus was forced to come to terms with another long-term injury.
United States winger Christian Pulisic threaded a pass through the Hamburg defence, which Batshuayi met at the far post for a tap in on 49 minutes.
Terodde's second goal did little for Cologne, seven points below Mainz in the relegation playoff place.
Valentino Lazaro fired Hertha ahead before the break on a counterattack, when Vladimir Darida played the ball through. Hamburg defended well and looked more like scoring.
The win left Dortmund third, a point behind second-placed RB Leipzig, who won 2-0 at home to Augsburg on Friday with Liverpool-bound Naby Keita hammering home a free-kick.