Central defender Martinez gave them a deserved lead in the 25th minute and fullback Bernat doubled it after the restart to make sure of Cologne's first home defeat in the Bundesliga since April 2016.
Bernat then set up Franck Ribery for their third goal in the final minute.
The Bavarians, who travel to Arsenal next week for their Champions League round of 16 second leg, are on 56 points with RB Leipzig, who drew 2-2 at Augsburg on Friday, in second on 49.
Borussia Dortmund, hosting Benfica on Wednesday for their own Champions League match, also made the most of Leipzig's slip-up, closing in to within six points in third place after Bundesliga top scorer Pierre-Emerick Aubameyang scored twice to help them demolish Bayer Leverkusen 6-2.
New VfL Wolfsburg coach Andries Jonker, who succeeded sacked Valerien Ismael this week, failed to inject any new life into the struggling former German champions, who drew 1-1 with Mainz 05.
(Reporting by Karolos Grohmann)
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