Scott Sinclair and Tom Rogic scored the goals in the first half.
Celtic, Scottish champions for the past six seasons, did not take up their allocation of tickets for the game because of security concerns, although some of their fans were present.
Their support is traditionally drawn from Glasgow's Catholic community whereas Linfield's is mainly Protestant.
The game was moved back two days to avoid clashing with anniversary celebrations of the 1690 Battle of the Boyne by Northern Ireland loyalists.
The second leg is in Glasgow next Wednesday and the winners will play Norwegian side Rosenborg or Ireland's Dundalk, who drew 1-1 in their first leg.
(Reporting by Steve Tongue, editing by Ed Osmond)
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