Trotters boss Gary Megson opted to leave Nicolas Anelka and El Hadji Diouf on the bench, with Swede Christian Wilhelmsson handed a start on the right flank.
The home side made the brighter start, using the long ball forward to Kevin Davies to good effect, however, Gavin McCann soon had to go off to be replaced by Mikel Alonso.
Chances were few and far between as the clock ticked on and midway through the first half Diogo Siston curled an effort narrowly wide of target from the edge of the area.
Just before the half-hour Giannakopoulos lifted a ball across goal from the right, but Davies couldn't get enough purchase on his header to trouble Konstantinos Chalkias.
A minute before the break Aris went in front as Nikos Karabelas' cross was headed away by Abdoulaye Meite, but only as far as Toni Calvo, who rifled home a low right-footer from 15 yards.
There was still time for Andy O'Brien to head against the crossbar from a corner as the game finally opened up.
The early minutes of the second half saw little goalmouth action, with Wanderers still looking short on ideas of how to open up the Greek side.
Just after the hour Giannakopoulos played in Kevin Nolan, but he sliced his shot wide, before Megson threw on both Anelka and Diouf.
Meite headed a Diouf free-kick narrowly wide soon after, as Bolton started to up the pace and 12 minutes from time Nolan drilled a right-foot shot inches too high from 20 yards out.
Five minutes later Nacho tried an audacious chip from fully 45 yards out and Jussi Jaaskelainen had to backtrack quickly to tip the ball over the crossbar.
Giannakopoulos then denied his compatriots a well-deserved victory with the last kick of the game, chesting down Nicky Hunt's long ball into the box and firing a low shot beyond Chalkias.