Hercules climb off the bottom of the La Liga standings with their first away win since September.
Djukic took over from the sacked Esteban Vigo just before the international break and inherited a side that had picked up just four points from a possible 33 since mid-January, but the former Yugoslavia international has made an immediate impact.
Javier Portillo gave Hercules the lead in the 70th minute and then Royston Drenthe, who this week was left out of training after arriving an hour late for the session, scored twice late on to make it 3-0 and seal the promoted club's first away win since stunning Barcelona at the start of the season.
It also meant the Alicante outfit moved up two places in the table, although they remain in the relegation zone.
Sociedad pulled a late goal back through Diego Ifran but it was not enough to save the freefalling Basque side, who have now lost five of their last six matches
Both teams - promoted from the Segunda Division last season - had chances in a goalless first half but it was not until 20 minutes from time when Hercules broke the deadlock with their first away goal since October.
The visitors did well in spreading the play from the left wing to the right, where Kiko Femenia curled in a brilliant cross that was headed home in the middle by Portillo.
On-loan Real Madrid player Drenthe then made it 2-0 in 84th minute, picking up Nelson Haedo Valdez's pass on the left, riding a challenge and then drilling a low shot into the far corner of the net.
Five minutes after that the Dutchman ensured the points would be heading back to Alicante when he made it 3-0, racing clear from the halfway line after a Sociedad attack had broken down before smashing a rising shot past Claudio Bravo.
Sociedad got on the scoresheet in the injury-time when Ifran side-footed in Xabi Prieto's low cross, but the game was already done and dusted by then as Hercules breathed fresh life into their hopes of avoiding the drop.