In Turkey’s last Group A game, Youssouf Ndayishimiye headed for a corner after just four minutes and, as the visitors came back into the game, they fell even further behind in the 33rd minute with a Serdar goal Gurler following a goalkeeper error. Craig Gordon.
Substitute Berkay Ozcan netted a 64th-minute third moments after coming on, with Hearts defender Nathaniel Atkinson scoring a 90th-minute consolation goal.
The home side, who won 4-0 at Tynecastle in September, finished top of Group A ahead of Fiorentina and both will go ahead.
With several key players still injured, it was a much changed Jambos side with youth that started the game and boss Neilson can be pleased with the aspects of the game and the experience gained.
Third-placed Jambos had no chance of progressing and Neilson, with his eye on Sunday’s cinch Premiership game with Motherwell, made a host of changes with Atkinson, Lewis Neilson, Jorge Grant, Cammy Devlin, Alan Forrest and Euan Henderson all entering.
Gordon equaled his fellow keeper’s records Henry Smith and ex-skipper Steven Pressley who played 22 games in European competition for the Jambos.
Basaksehir, meanwhile, showed two changes to the team that won in Gorgie with Junior Caicara and Ndayishimiye starting and it was the latter who gave the home side an early lead.
Stefano Okaka slipped a shot past the 12-yard post after just 90 seconds, but the Edinburgh side ignored the warning and soon after Ndayishimiye had plenty of time and space to he needed to overtake the helpless Gordon from a Deniz Turuc corner.
In a quick and impressive response, Hearts piled forward and Henderson came over from 20 yards and then put the home side under pressure all over the pitch.
However, just after the half hour mark, Gordon rushed from his goal to save Okaka from finding himself on the end of a long ball from the defense but failed to connect properly and Gurler took his time before to shoot into the net in front of defender Toby Sibbick, who had retired. at the line.
As the visitors faltered Gordon made a double save from Bertrand Traore but the Tynecastle side ended the first half holding on.
The Turkish side started the second half well, however, and in the 56th minute Ndayishimiye got another free header from a corner, but this time his effort flew wide.
Neilson brought in Josh Ginnelly, Barrie McKay and Lawrence Shankland for Forrest, Henderson and Connor Smith just after the hour mark.
However, moments later, when defender Alex Cochrane lost possession of a Gordon throw-in, Omer Ali Sahiner squared for Ozcan, just on the pitch, and he sorted his feet before hammering the ball high in the net.
Hearts kept trying to grab a goal to give their traveling fans something to cheer for and they got it in the 90th minute when goalkeeper Muhammed Sengezer parried a shot from Shankland with Atkinson within reach to send the ball in the net.