Nuno Gomes had headed Rovers in front early on, before Best doubled the lead with a fantastic left-foot finish.
But his involvement ended before the break, as he went down in agony clutching his knee, although it was hoped the injury was not as bad as first feared.
The game never really got going in the second half, as both sides rang the changes, but the vociferous Greek fans did have something to cheer about late on when substitute Nikolaos Kourelas got in behind the Rovers defence to score.
In glorious sunshine, Rovers started the game brightly, with Danny Murphy pulling the strings from a deep-lying midfield position.
And it was from one of the skipper's pin-point passes that Rovers opened the scoring in the seventh minute. Marcus Olsson picked up possession wide on the left, his cross was headed goalwards by Best and Gomes glanced it in.
Rovers came close to doubling their lead from a corner on 11 minutes. Murphy's cross, at the second attempt, was headed against the bar by Gael Givet and Best smashed home the rebound, but the former Newcastle frontman was adjudged to be offside.
Athens' best chance of the first half arrived on 16 minutes when Enias Kalogeris slipped in Thomas Tsitas, but Grant Hanley got back to make the block.
Steve Kean's men did double their advantage just before the half-hour mark with a goal of real quality. Murphy switched the play with another sumptuous pass to Best, who brought the ball down on his chest, cut inside onto his left foot and curled a sublime shot into the far top corner.
But the half ended on a sour note for the Irishman, who went to ground under an innoculous challenge from Yago Fernandez, which resulted in Best being stretchered off. Mauro Formica coming on as his replacement.
Rovers made a raft of changes at the break, with David Dunn, Jason Lowe, Scott Dann, Adam Henley and Jake Kean all entering the fray.
And it was one of the substitutes who almost increased Rovers' lead on 57 minutes when Formica back-heeled the ball into the path of Dunn, whose long-range strike sailed just past the upright. At the other end, an Antonios Stavros Rikka free-kick missed by a similar margin.
Just past the hour mark, Rovers had a lucky escape when Jake Kean passed the ball straight to Panagiotis Lagos, who lobbed the ball into the side-net with the goal gaping.
With the game in need of a spark, Rovers brought on exciting Portuguese teenage duo Fabio Nunes and Edinho, but it was the tourists who grabbed the next goal on 76 minutes. The Greeks slipped the ball in between Henley and Hanley, and substitute Nikolaos Kourelas raced through to slide a left-foot shot under Jake Kean and into the far corner of the goal.
Rovers almost responded immediately when some silky skill from Formica won a free-kick, which was fired just wide by Dunn.
After back-to-back defeats, it was a welcome pre-season win for Rovers, who head off to Holland next for a Dutch double-header.
ROVERS: Robinson (Kean 46), Orr (Edinho 72), Hanley, Givet (Dann 46), Morris, Ribeiro (Henley 46), Murphy (Lowe 46), Jorge, M. Olsson (Nunes 72), Gomes (Dunn 46), Best (Formica 44).
Subs not used: Petrovic, Rochina, Goodwillie.
AEK ATHENS: Arampatzis (Konstantopoulos 57), Arkoudas, Bougaidis (Nikolias 84), Tsamourlidis (Vlachos 80), Lagos (Tomaras 66), Rikka (Fetsis 80), Fernandez (Matenzidis 80), Papadimitriou (Grontis 32), Kalogeris (Fountas 46), Guerreiro (Kourelas 59), Tsitas (Stamatis 32).
Subs not used: Kostaridis, Agyiriba, Nikoltsis.