Heiko Westermann scored an 82nd-minute goal as the Bundesliga strugglers prevailed 2-1 at home to Leverkusen to climb out of the relegation zone.
Hamburg are now one point clear of Nuremberg and the relegation play-off spot after snapping their three-game winless streak.
"It's a great feeling for the lads," Slomka said.
"This win isn't just good for the team, it's good for the city, for the fans. I think everyone deserves it."
Veteran German defender Westermann was the unlikely hero for Hamburg.
The 30-year-old made a timely run into the penalty area before latching onto a Dennis Diekmeier cross with a well-executed volley.
It was Westermann's second goal of the season and a relief for Hamburg, who went into the match with just two league wins since the New Year.
"It's a relief," said Westermann.
"So many emotions in the game. Of course it's a great feeling when you score the winning goal."
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