A tight 90 minutes produced few clear chances for either side, and the winner will be determined when the pair lock horns in the second leg at Turf Moor next week.
As for Burnley, Joe Hart made his debut following a move from Manchester City, while fellow new arrival Ben Gibson started on the bench. Matej Vydra, however, had not been cleared to feature for his new club.
The Premier League outfit had to wait until the 28th minute to have their first clear sight of the home side's goal, but James Tarkowski failed to make the most of a knockdown from fellow centre-back Ben Mee.
Hart's first serious work in a Burnley shirt arrived in the 36th minute when he was forced to tip a Manuel da Costa header over, and the two teams entered the half-time interval at 0-0 following a tight first 45 minutes.
Basaksehir had a super chance to make the breakthrough just past the hour when Milos Jojic found time and space inside the box, but the substitute could only hit the side-netting.
The fact that Hart was booked for time wasting in the 69th minute was an indication of Burnley's approach, with the Premier League outfit seemingly happy to take a 0-0 draw back to Turf Moor.
The Turks were the team pressing for a goal as the match entered four minutes of additional time, but neither side could make the breakthrough as it finished goalless in Istanbul.
ISTANBUL BASAKSEHIR (4-2-3-1): Gunok; Caicara, Da Costa, Epurenau, Clichy; Belozoglu, Tekdemir; Elia (Frei 84'), Kahveci (Jojic 58'), Visca; Bajic
BURNLEY (4-3-3): Hart; Bardsley, Tarkowski, Mee, Taylor; Hendrick, Westwood, Cork; Barnes (Lennon 78'), Walters, Gudmundsson (Vokes 62')