It was also the first home win since March for Hamburg, who had drawn their previous three games under Fink following his move from Swiss champions FC Basel.
"We knew it was a question of time as we had been playing well," said Hamburg's sporting director Frank Arnesen.
"It's a big relief for us and the fans," added captain Heiko Westermann.
Michael Oenning quit after their poor start while interim coach Rodolfo Cardoso's provisional licence to lead a Bundesliga club expired after two league games in charge, leading Arnesen to take over for one game.
Hamburg, unbeaten in their last five games, climbed to 14th with 13 points from 13 games while Hoffenheim are ninth with 17.
Peru striker Guerrero broke the deadlock in the 25th minute, scoring at the second attempt from Andreas Beck's pass after his first shot hit the post.
Jansen burst into the penalty area, leaving three defenders in his wake, and found a gap between Tom Starke and the near post to extend Hamburg's lead in the 65th minute.
Stuttgart, sixth with 21 points, and Augsburg produced an uninspiring first half until Austria forward Harnik rifled home an impressive volley in the 41st minute after the visitors had failed to clear a throw-win.
The visitors barely had time to celebrate before Harnik produced a superb effort, bursting down the left and then cutting inside to curl his shot just inside the far corner.
Rafati was reported as out of danger and in a stable condition following the incident.
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