Robin van Persie got the Gunners off to the perfect start when he opened the scoring after eight minutes, before Adebayor then took centre stage with a lethal treble as Arsene Wenger's side picked up their first away points of the new season.
In contrast, there was precious little to cheer for Wenger's opposite number, Paul Ince, whose side shipped four goals for the second successive game.
Ince made two changes to the team that lost 4-1 at West Ham a fortnight ago with Danny Simpson and Keith Andrews both coming in for their home league debuts in place of the injured Andre Ooijer and Vince Grella.
Meanwhile, England hat-trick hero Theo Walcott started on the left-hand side of Arsenal's midfield, fresh from his stunning exploits against Croatia in midweek.
It took the speedy winger just eight minutes to pick up from where he left off in Zagreb, leaving a couple of Rovers players trailing in his wake when he embarked on a mazy dribble across the edge of the penalty box.
He then coolly fed the unmarked van Persie, who prodded a low shot beyond the sprawling Paul Robinson from six yards out to give the visitors the lead.
Rovers almost restored parity two minutes later when Roque Santa Cruz curled in a left-footed shot that looked destined for the back of the net until Kolo Toure somehow headed it over for a corner.
However, Arsenal always looked threatening on the break and they should have doubled their advantage when Adebayor sent Walcott scampering clear, but for once the England man fluffed his lines.
Rovers created several promising openings at the opposite end, but they lacked the decisive touch needed to take full advantage.
Then right on the stroke of half-time, Arsenal inflicted a major psychological blow when they conjured a killer second goal.
Brazilian midfielder Denilson was afforded far too much time and space to deliver a cross from the left and Adebayor rose unchallenged to send a powerful downward header past Robinson.
The Gunners continued to turn the screw after the re-start with Toure and Emmanuel Eboue both bringing the best out of Robinson with two efforts in the space of 30 seconds.
Arsenal grabbed a third goal in the 81st minute when Adebayor confidently tucked away a penalty following Stephen Warnock's foul on Eboue.
The in-form Gunners striker then completed his hat-trick deep in injury time after skilfully rounding Robinson when sent clear by Denilson.