With the hosts already leading 2-0 thanks to goals from Petr Jiracek and Pierre-Michel Lasogga, Calhanoglu lined up a 50-yard free-kick in stoppage time and sent an incredible dipping, swerving effort beyond Roman Weidenfeller into the top corner.
It capped a perfect start to the new era at Hamburg, Slomka having taken over from Bert van Marwijk after the Dutchman had overseen a run of eight consecutive defeats in all competitions.
A first-half header from Jiracek and Lasogga's effort after the break put Hamburg on the way to a first league win since November 24, before Calhanoglu's stunning strike gave the strugglers a timely boost in front of their home fans.
The result was a rude awakening for Dortmund, who came into the game on the back of four straight wins in all competitions and is a setback ahead of their UEFA Champions League tie with Zenit on Tuesday.
Slomka rung the changes from Van Marwijk's last XI, who had lost at Eintracht Braunschweig, making six alterations with Rafael van der Vaart among those to miss out, though his absence was forced through torn ankle ligaments.
After a cagey opening, Jiracek had the first sight of goal in the 16th minute, cutting inside Lukasz Piszczek on the edge of the box before curling a tame shot straight into the arms of Roman Weidenfeller.
Henrikh Mkhitaryan saw a 25-yard effort deflected narrowly wide two minutes later, before Robert Lewandowski clipped the ball over the crossbar after being picked out by Marcel Schmelzer.
But it was the previously beleaguered hosts who went in front with half-time approaching as Lasogga somehow dug out a cross from the byline, and Jiracek stole in at the back post to head home.
Jurgen Klopp introduced Marco Reus at the break in a bid to provide Dortmund with greater penetration and the Germany international was involved immediately, sending an effort two yards wide within five minutes of the interval.
Just before the hour, Lasogga doubled the advantage with his 11th goal of the season.
Calhanoglu robbed Nuri Sahin in the middle of the park, and Tolgay Arslan slipped the ball through to Lasogga, who ran clear on goal and coolly slotted beyond Weidenfeller.
Rene Adler maintained the two-goal cushion, tipping away Pierre-Emerick Aubameyang's volley before Heiko Westermann saw his header ruled out for a Slobodan Rajkovic push at a corner.
Adler had to be alert again in the 70th minute beating away substitute Marvin Ducksch's fierce drive, 60 seconds before brilliantly denying Lukasz Piszczek's header.
Calhanoglu's wonder effort, that swerved several directions to deceive Weidenfeller capped a marvellous outing for the hosts, as they picked up a vital boost in their bid to avoid a first-ever relegation from the Bundesliga.
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