Mancini struck in the 27th minute when Sweden striker Zlatan Ibrahimovic [ Images ] held off two defenders to slide the ball through to him on the left-hand side of the box.
Inter did not have it all their own way as Bryce Moon [ Images ] hit Julio Cesar's [ Images ] crossbar and the Greek side missed several other opportunities to level at the Olympic [ Images ] Stadium in Athens.
The home side dominated the second half but were undone five minutes from time when substitute Adriano latched on to a through ball from Ibrahimovic and lashed the ball past Panathinaikos goalkeeper Mario Galinovic.
Inter had the best of the first half with Patrick Vieira edging Gilberto Silva [ Images ] in the midfield battle between two ex-Arsenal team mates.
Ibrahimovic and Ricardo Quaresma both forced fine saves from Galinovic but Inter could not find a way to increase their lead.
After the break the Italian champions went off the boil as Gilberto Silva took control of the midfield. Panathinaikos created numerous chances, the best of them falling to Cleyton who narrowly missed twice.
While Inter turned to their substitutes bench bringing on Luis Figo, Sulley Muntari and Adriano, Panathinaikos began to wilt and Adriano finished things off in the 85th minute.