Melbourne Victory made a blistering start to the contest and took an early lead when Berisha scored from the penalty spot inside 15 minutes.
Bozanic then added to the advantage of the half hour when he curled in a delightful free-kick that got the better of Eugene Galekovic.
A late Marcelo Carrusca proved merely consilation for Adelaide.
In front of over 23,415 at Etihad Stadium, Victory were on the front foot immediately against their fierce rivals and Berisha almost scored one of the quickest goal's in the comeptition's history.
It what was an open start to the contest, Adelaide too were looking threatening, in particular Spanish winger Sergio Cirio.
Yet it was Victory that had the first great opportunity to open the scoring. After a flowing move, Kosta Barbrouses slipped a pass through for Jason Geria in the box, who was brought down after his first touch took him past a defender and the hosts' were awarded a penalty.
Berisha sent Galekovic the wrong way from the spot to make it 1-0 and set a new A-League record of scoring in six consecutive home games.
The Reds came close to providing an immediate reply as a Craig Goodwin free-kick near the byline forced Danny Vukovic into a difficult save to keep it out.
Aside from that scare, Melbourne were in complete control of the contest and doubled the lead with a well-placed Bozanic free-kick from the edge of the box on the half hour.
Shortly after the goal, Victory fans in the North Terrace staged their protest against the treatment of A-League supporters and left the ground, while continuing to sing outside the stadium.
There was little sign that Adelaide could stage a comeback in the second half. In fact it was the home side that looked more likely to add to the score, with Galekovic at full stretch to keep Fahid Ben Khalfallah.
Adelaide did give themselves some hope of a late fightback when Carrusca scored a late penalty, but it failed to inspire the Reds further.
Victory coach Kevin Muscat said he was disappointed at his side's second-half drop in intensity.
"I don't think Adelaide ran over us - I think we made it a contest," he said.
Adelaide boss Guillermo Amor said he felt his side deserved a share of the points.
"It is a good sign when everyone in the stadium is waiting for a final whistle. It shows what sort of match it has been," he said through a translator.