The encounter at Dortmund's Signal-Iduna Park was marred by the death of a fan, who fell from a stadium stand before the match and later died in hospital. Police said the fall was most likely an accident.
Leverkusen had little trouble overcoming relegation-threatened Cottbus, scoring three times in the first half and showing why they are fifth in the league, only three points from the top.
Patrick Helmes, the team's top scorer with 12 goals this season, struck after 12 minutes when Arturo Vidal headed a Tranquillo Barnetta cross from the right into the path of the striker who had no problem firing past keeper Gerhard Tremmel.
Defender Michal Kadlec grabbed a second in the 29th with a powerful header before Renato Augusto added a third two minutes before the break with a solo run and an unstoppable shot from 25 metres into the right corner.
Cottbus were lucky not to concede more goals before Leverkusen took their foot off the gas in the second half and allowed the visitors to score in stoppage time through Ervin Skela's well-struck free kick.
WERDER COMEBACK
Werder Bremen, who last year lost to Dortmund at the same stage of the Cup, had more than 70 percent of the possession but it was the hosts who struck first after 11 minutes when Mohamed Zidan beautifully timed his pass for Alexander Frei to slot past Bremen keeper Tim Wiese.
Bremen, hugely inconsistent this season, grabbed a deserved equaliser in the 62nd with Hugo Almeida, who fired from the edge of the box into the left corner.
With Dortmund defending deep, Bremen pushed forward and Claudio Pizarro scored the winner 10 minutes from time when a Diego free kick fell into the path of the Peruvian who fired home from five metres.
Wehen Wiesbaden, lying deep in the second division's relegation zone, caused a minor upset, beating Bundesliga team and hosts Karlsruhe 1-0 thanks to a Ronny Koenig strike in the 73rd.
Wolfsburg were made to fight for 58 minutes before breaking the resistance of second-division Hansa Rostock, eventually winning 5-1 with a hat-trick from Brazilian Grafite.
Bayern Munich, Hamburg SV, Schalke and Mainz have already qualified, with the quarter-final draw set for Sunday.
