Burnley was without key players including Chris Wood, Steven Defour and Robbie Brady for the trip to Turkey ahead of the start of the Premier League season.
Although Basaksehir had the better chances in a tight match at the Fatih Terim Stadium, Burnley held on to prevent the host taking a lead into next week's return leg at Turf Moor.
The winner of the tie will face either Olympiacos or Luzern for a place in the group stages, with the Greek side leading 4-0 after the opening leg.
Former Turkey international Emre pulled the strings in the Basaksehir midfield but the host lacked end product in the final third during the opening exchanges.
Hart, making his first appearance after joining from Manchester City, had to push away Junior Caicara's cross-turned-shot, with the goalkeeper then turning over a header from Manuel da Costa in the 36th minute.
James Tarkowski and Jonathan Walters both failed to bundle home in the box as Burnley made a strong start to the second half.
Edin Visca proved a handful for the Burnley defence and when the striker flicked on a long pass, substitute Milos Jojic should have scored but hit the side netting at Hart's near post.
That miss proved the decisive moment in the tie's first leg, as referee Srdjan Jovanovic waved away late penalty appeals for a Sam Vokes handball, with Basaksehir travelling to Lancashire for next Friday's (NZST) reverse fixture.