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.
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.
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.
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.
