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.
Martin Harnik scored twice, including a superb solo effort, to give VfB Stuttgart a 2-1 win over bottom-of-the-table Augsburg in Sunday's other match.
"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.
Paulo Guerrero and Marcell Jansen scored in each half for Hamburg where Fink is the fourth man to occupy the hot seat this season.
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.
Promoted Augsburg levelled almost immediately after the re-start when Daniel Baier pulled the ball back to Tobias Werner who side-footed into net from the edge of the penalty area.
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.
Bayern Munich lead with 28 points from 13 games after their 1-0 defeat on Saturday to Borussia Dortmund, who are joint second with Borussia Monchengladbach two points behind the Bavarians.
The weekend was overshadowed by the attempted suicide of referee Babak Rafati hours before he was due to take charge of Saturday's game between Cologne and Mainz 05, which was postponed.
Rafati was reported as out of danger and in a stable condition following the incident.
‘After Manchester City’s recent form, maybe they’re the underdogs against Manchester United!’ Former Red Devils defender on this weekend’s derby
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
