Djite's first-half goal at Adelaide's Coopers Stadium lifted the Reds to seventh on the ladder, just a win behind sixth-placed Central Coast.
Newcastle remain fourth after a match which boiled in the second half when four yellow cards were dished out in an ill-tempered 16 minutes.
Djite scored in the 26th minute after his influential Spanish teammate Cirio produced a slicing midfield run and passed to Jeronimo Neumann, who angled a dangerous cross which Jets goalkeeper Mark Birighitti did well to cut off.
In the ensuing scramble, the ball deflected to an unmarked Djite who slotted into an unguarded goal.
Adelaide controlled the opening half as the visitors failed to muster a shot on target - and were fortunate not to have Taylor Regan marched for an ungainly tackle which felled Adelaide playmaker Marcelo Carrusca.
Regan was fortunate to only receive a yellow card and Adelaide's captain Eugene Galekovic, coach Josep Gombau and Carrusca all approached referee Chris Beath at halftime to express their anger.
The heat extended into the second half of a game watched by 10,778 spectators as Carrusca and his marker, Jets captain Ruben Dadkovich, engaged in a running battle.
Moments after a plea for calm from referee Beath, Zadkovich brought down Carrusca and was yellow-carded.
Three minutes later, Adelaide's Jeronimo was booked for a similar take-down of the Jets' skipper.
And two minutes on, Carrusca sought personal revenge with a menacing studs-up tackle on Zadkovich which didn't connect but earnt a booking.
Newcastle's Nathan Burns was yellow-carded soon after for another crude tackle on Carrusca, who was substituted after his repeated batterings.
The spite came as the Jets failed to convert their best chance to equalise when Adam Taggart closed on goal in a one-on-one with Galekovic, only for his shot to be blocked by the keeper's leg.
Adelaide substitute Jake Barker-Daish went nearest to a second-half score when a long-range shot in the 88th minute managed to hit both side posts but didn't cross the goal-line.
Adelaide's coach Gombau was disappointed in the treatment dished out to Carrusca, who will miss Friday night's match against Sydney FC with leg bruising.
"I am disappointed in this because we need to protect the players, we need to protect clean football," Gombau said.
"But I cannot do nothing. The referee's job is very difficult, but we need to protect the players, it's important."
But Newcastle coach Gary van Egmond said his Jets weren't the only side with a physical intent.
"He (Gombau) is entitled to his opinion," he said.
"There were some poor challenges on both sides. The one that Carrusca did on Ruben, although he (Zadkovich) didn't roll around 15 times, was a poor challenge.
"It's fair to say both sides were fairly physical."