Firmino put the visitors in front after just five minutes with Rafael van der Vaart levelling from the penalty spot just before half-time.
Kevin Volland gave Hoffenheim the lead again early in the second half before Anthony Modeste grabbed a brace and Firmino supplied the finishing touch to an emphatic win for Markus Gisdol's men.
Hoffenheim made a flying start to their season against Nurnberg a week ago, racing into a two-goal lead before being pegged back for a draw, and they were in front after only five minutes in Hamburg.
Firmino beat Heiko Westermann to Fabian Johnson's cross to head past Rene Adler from close range with the HSV-Arena falling silent.
Hamburg forward Hakan Calhanoglu tested Koen Casteels with an overhead kick before Maximilian Beister wasted a good chance with only Casteels to beat - opting to pass to a team-mate who was not there rather than shoot - as the northern Germany side tried to hit back.
Hamburg's pressure finally paid off a minute before half-time when Tarik Elyounoussi handled the ball inside the penalty area and Van der Vaart swept in the resulting spot-kick.
Hoffenheim were quick out of the blocks once again in the second half, re-taking the lead after just six minutes through Volland, who had a goal disallowed last weekend.
Firmino was the provider this time, sending Volland through with a perfectly-timed pass which meant the forward only had to think about where to place his shot, which he did decisively.
Modeste doubled the visitors' lead midway through the second half with Firmino again bagging the assist with a cross from the right which Hoffenheim's new recruit turned in.
Hamburg conceded three goals at Schalke a week ago and still came away with a point, but this time, things were to get worse for Thorsten Fink's men.
Modeste added his second of the game and Hoffenheim's fourth in the 75th minute with the very same combination as for the visitors' third goal before Firmino got his name on the scoresheet again to round off victory.