Last season's runners-up Leverkusen, who are all but eliminated from the Champions League after a 3-1 home loss to Barcelona in their round of 16 first leg, were under growing pressure after just two wins in their last seven league games.
Germany midfielder Bender gave Dutt something to smile about, putting the visitors ahead in the 16th minute when he slotted in from a metre out after a Vedran Corluka header.
The talented 22-year-old then made sure of victory early in the second half, picking up a brilliant Renato Augusto pass and rifling in with his left foot as goalkeeper Michael Rensing charged forward.
"It is very important to win a derby and even more important to build on our win against Augsburg last week," Bender told reporters. "We fully deserved the win today."
Cologne were toothless throughout despite the return of their top scorer Lukas Podolski following an ankle injury in late January.
In other action, Nuremberg stunned hosts Werder Bremen courtesy of a 65th-minute Alexander Esswein strike against the run of play to leave Thomas Schaaf's team in sixth place on 36 points and without a home win since mid-December. Nuremberg move up to 10th on 28.
Augsburg's Torsten Oehrl struck twice in two minutes to help his relegation-threatened team to a 3-0 win over troubled Hertha Berlin and spoil veteran coach Otto Rehhagel's Bundesliga comeback.
Rehhagel, 73, the former long-time Werder Bremen coach who won Euro 2004 with Greece and returned to the Bundesliga after 12 years, could do nothing to avoid Hertha's seventh consecutive defeat in all competitions that sent them down to 16th spot.
"We are now in the midst of a relegation fight and that means we have to fight," Rehhagel, who has been involved in a record 1,022 Bundesliga games as a player and a coach, told reporters. "We really need to wake up and face that."
Mainz 05 crushed lowly Kaiserslautern 4-0 to record their first win after three successive draws and move up to 11th on 27 points while VfB Stuttgart beat Freiburg 4-1 to go up to eighth.
Borussia Monchengladbach drew 1-1 at home to Hamburg SV on Friday.
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