Slomka, appointed as Hamburg's third coach of the season on Monday, saw his side produce an excellent performance to end a run of a five straight defeats with a 3-0 victory - with the final goal in particular a contender for the strike of the season.
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Czech midfielder Petr Jiracek opened the scoring for the hosts three minutes before the interval as he headed home a cross from Pierre-Michel Lasogga, before Lasogga then doubled Hamburg's lead with a cool finish of his own on 58 minutes.
But the undoubted highlight of the game was a stunning strike from Hakan Calhanoglu late on, as he capped off a fine display with a magnificent 50-yard free-kick that swerved beyond Roman Weidenfeller and into the top corner.
Defeat for Dortmund gave second-placed Bayer Leverkusen the chance to open up a four-point lead over Jurgen Klopp's men, but Sami Hyypia's side missed that opportunity by falling to a 3-1 loss at Wolfsburg.
A strong second-half showing secured all three points for the hosts, with goals from Luiz Gustavo and Ricardo Rodriguez moving Wolfsburg within two points of the top four after Sidney Sam had cancelled out Bas Dost's opener.
Freiburg dropped into the automatic relegation spots courtesy of a 4-2 home defeat to Augsburg.
The hosts looked to be on their way to a rare triumph after Jonathan Schmid and Admir Mehmedi had turned things around following Tobias Werner's seventh-minute opener for Augsburg.
But an enthralling encounter was settled in dramatic fashion as late goals in a 12-minute spell from Paul Verhaegh, Halil Altintop and Andre Hahn settled the game to keep Augsburg in contention for a place in the Europa League.
Meanwhile, 10-man Nuremberg beat Eintracht Braunschweig 2-1, with quickfire second-half strikes from Hiroshi Kiyotake and Tomas Pekhart giving the hosts the points after Dominick Kumbela had capitalised on Per Nilsson's dismissal to give Braunschweig the lead.
An 82nd-minute Sejad Salihovic penalty earned Hoffenheim a 2-2 draw at Borussia Monchengladbach, the midfielder scoring from the spot to complete the fightback after Roberto Firmino had reduced the deficit following goals from Patrick Herrmann and Tony Jantschke.
Elsewhere, European hopefuls Hertha Berlin won 2-1 at struggling Stuttgart, forward Sandro Wagner scoring the winner after 87 minutes before being shown a red card in injury time for a second bookable offence.