Poland's Robert Lewandowski struck the winner just before the break with Cologne goalkeeper Michael Rensing making half a dozen sensational saves and the hosts rattling the bar twice.
Dortmund's 19th win in 25 games lifted them to 61 points before second-placed Bayer Leverkusen welcome VfL Wolfsburg on Saturday.
"This must be one of the most deserved 1-0 wins ever," said visibly relieved Dortmund coach Jurgen Klopp. "My players worked non-stop over 90 minutes and we created many chances."
Cologne felt the full force of Dortmund's formidable attack early on but Rensing, who joined the club on a free transfer this season after leaving Bayern, was in superb form and kept the hosts at bay several times with spectacular saves.
Dortmund's pressure finally paid off when Lewandowski picked up the ball outside the box and scored with a powerful drive which went in off the post.
