Anthony Modeste was ruled to have been brought down by Bosnian central defender Emir Spahic, who was sent off with a red card, but television replays showed he may stumbled over the ball rather than fouled by his opponent.
The Frenchman, however, kept his nerve and converted his 81st-minute spot- kick to lift the hosts to seven points from three games, leaving Hamburg coach Bruno Labbadia fuming on the sidelines.
Hamburg, the only team to have played in the Bundesliga every year since the top league’s creation in 1963 but who needed a relegation playoff to stay up last season, had grabbed the lead through Lewis Holtby two minutes after the restart but Cologne levelled through Philipp Hosiner in the 76th.
Former German champions VfB Stuttgart, who also narrowly stayed up last season, lost their third straight game, slumping to a 4-1 defeat at Eintracht Frankfurt, piling early pressure on new coach Alexander Zorniger whose team have yet to win a point.
To make matters worse Stuttgart were left with 10 men when keeper Przemyslaw Tyton was sent off with a straight red card in the 67th minute.
Promoted Ingolstadt continued their fine start to their maiden Bundesliga season with their second win of the campaign, beating fellow Bavarians Augsburg 1-0 courtesy of Australia international Mathew Leckie’s 63rd-minute long-range effort.
Champions Bayern Munich face Bayer Leverkusen later on Saturday, with both teams on six points, one behind VfL Wolfsburg who beat Schalke 04 3-0 on Friday.
(Reporting by Karolos Grohmann, editing by Ed Osmond)