Pierre-Michel Lasogga put Hamburg ahead with their first goal of 2014, but two Kumbela goals turned the game around.
Ivo Ilicevic brought Hamburg on terms once again, but Kumbela completed his hat-trick before Jan Hochscheidt sealed victory for Torsten Lieberknecht's men, who are now just a point adrift of Hamburg at the foot of the table.
Felix Magath may no longer be looming to take Van Marwijk's place - the veteran coach took the Fulham job on Friday - but that did not mean his position was any safer as Hamburg sought to avoid a seventh straight league defeat.
After an even start, the visitors took the lead in rather bizarre fashion.
Marcel Jansen sent in a long throw-in which was headed on by Lasogga, who had his back to goal and appeared merely to be helping the ball on into the danger area.
However, it looped over goalkeeper Daniel Divari and nestled in the far corner of the Braunschweig goal.
It was the stroke of luck Hamburg needed to score their first goal of 2014, and it appeared to give them a psychological boost too.
Rafael van der Vaart went close with a direct free-kick which Davari managed to push away at the last minute before Ilicevic wasted a good chance in the second half when he completely missed the ball from Ola John's cross.
It was a costly miss as, moments later, Braunschweig levelled through Kumbela.
Karim Bellarabi's cross was flapped at by Rene Adler with the ball falling to the Braunschweig striker, who kept his cool to bag his third goal of the season.
His fourth followed on the hour mark when he was quickest to meet Mirko Boland's cross to score from close range.
Kumbela, who had only entered the action as a half-time substitute, almost completed his hat-trick to make it three for Braunschweig, but Adler made a fine save to deny him and keep Hamburg in the game.
Hamburg fought back defiantly and levelled through Ilicevic's fine shot with the outside of his boot in the 76th minute.
That goal stung Braunschweig, who responded with Kumbela's third goal after Adler did all he could to keep two shots out.
Bellarabi nearly sealed things in stoppage time before Hochscheidt completed the victory on the next counter-attack for the hosts.