Two goals from Mike Hanke and one from Juan Arango secured a fourth straight league win that saw them move on to 29 points.
"This was a deserved win," Gladbach coach Lucien Favre told reporters after his team stormed into a 2-0 lead at the break before adding a third just after the interval. "We could have gone into the second half with an even bigger lead.
"We wanted to pressure them from the start and my players did it well. We could have won by four or five goals today," said the Swiss, who took charge midway through last season to keep them in the Bundesliga.
Gladbach, who last season needed a relegation play-off to stay up, sent in front when forward Hanke fired in a low drive from 14 metres in the 20th minute.
Venezuela international Arango doubled the lead on the half hour, curling a 23-metre free-kick over the wall past goalkeeper Michael Rensing as Gladbach found little resistance in the regional derby that saw crowd trouble before the start.
Hanke dashed any Cologne hopes of a second-half comeback by slotting the ball home in the 47th after Rensing had palmed away a Patrick Herrmann shot.
