Corentin Tolisso fired in the winner in the 52nd minute after some sloppy defending from 10-man Hamburg to lift them to 20 points, level with Dortmund who let a two-goal lead slip to draw 2-2 at Eintracht Frankfurt.
Bayern, who twice hit the post, found it hard going against a spirited Hamburg side but once Gideon Jung was sent off with a straight red card in the 38th minute, the visitors gradually found more space.
Tolisso snatched the winner seven minutes after the restart as Heynckes, who took charge earlier this month after Carlo Ancelotti was sacked, made it three wins in three games in all competitions since taking over.
"You could see Hamburg were very motivated and were fighting hard," Heynckes told reporters. "We struggled from the start and we did not keep the tempo up. We did it a bit better after the break and had a lot of chances but those must go in.
"At 2-0 the game is finished. Instead we had to be nervous until the end."
Dortmund squandered a two-goal lead and have now failed to win any of their last three matches in all competitions.
Midfielder Nuri Sahin took over as centre forward to slot the ball for the lead in the 19th minute.
Bundesliga top scorer Pierre-Emerick Aubameyang should have added another in the 51st, but his shot from 10 metres flew over the bar.
The 23-year-old Maximilian Philipp made amends a little later, underlining his status as one of the rising stars in German football with his fifth goal of the season.
Dortmund looked on track for a comfortable win after losing to RB Leipzig last week and then going on to draw at APOEL Nicosia in the Champions League, but Frankfurt turned the game around in four minutes.
Sebastien Haller converted a penalty in the 64th minute and Marius Wolf brought the hosts level when he fired in off the post in the 68th as Dortmund were left stunned by the rapid turnaround.
Leipzig moved up to 19 points in third place, after edging past VfB Stuttgart 1-0.
Bayer Leverkusen fell a goal behind but then scored five times with five different players in the second half to crush Borussia Moenchengladbach 5-1 and climb to ninth on 12 points.
(This story corrects paragraph four to make clear Heynckes took charge of Bayern in October and not last month.)
(Reporting by Karolos Grohmann; Editing by Christian Radnedge and Pritha Sarkar)
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