Cetlic - playing in Edinburgh because of Parkhead's use in the Commonwealth Games - were cheered on by a 35,000 'home' crowd but they were unable to inspire Ronny Deila's side.
The Scottish champions dominated early possession without creating any clear-cut openings although Adam Matthews had a penalty appeal turned down when he went down in the box having run past three Legia defenders.
But on 36 minutes Legia extended their lead in the tie when Michal Zyro, who scored in last week's first leg, latched on to a long ball over the top before coolly slotting past Fraser Forster.
The second half followed a similar pattern to the first with Celtic trying to take the game to their opponents, Anthony Stokes carving out an opening after 55 minutes but seeing his curled effort from the edge of the area comfortably saved by Dusan Kuciak.
And just six minutes later Legia killed off Celtic's hopes of a comeback after capitalising on more sloppy defending.
This time Michal Kucharczyk got on the end of a delightful throughball from Miroslav Radovic before effortlessly rounding Forster and slotting the ball home.
Callum McGregor and Leigh Griffiths had late efforts for Celtic but were unable to trouble Kuciak, leaving them with only a Europa League play-off tie to look forward to.