In the final Group A fixture in Turkey, Youssouf Ndayishimiye headed in a corner after just four minutes and, while the visitors played their way back into the game, they fell further behind in the 33rd minute to a Serdar Gurler goal following a mistake by keeper Craig Gordon.
Substitute Berkay Ozcan drove in a third in the 64th minute moments after coming on, with Hearts defender Nathaniel Atkinson scoring a consolation goal in the 90th minute.
The home side, who had won 4-0 at Tynecastle in September, finished top of Group A above Fiorentina and both will go forward.
With several key players still out injured, it was a much-changed Jambos side laced with youth which started the game and boss Neilson can be happy with aspects of the match and the experience gained.
The third-placed Jambos had no chance of progressing and Neilson, with one 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 coming in.
Gordon matched the records of fellow goalkeeper Henry Smith and ex-skipper Steven Pressley of playing 22 games in European competition for the Jambos.
Basaksehir, meanwhile, showed two changes to the side 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 slid a shot past the post from 12 yards after only 90 seconds but the Edinburgh side failed to heed the warning and soon afterwards Ndayishimiye had all the time and space he needed to head past the helpless Gordon from a Deniz Turuc corner.
In a swift and impressive response, Hearts piled forward and Henderson drove over from 20 yards and subsequently put the home side under pressure all over the pitch.
However, just after the half-hour mark Gordon rushed from his goal to prevent Okaka getting on the end of a long ball from defence but failed to connect properly and Gurler took his time before firing into the net past defender Toby Sibbick, who had retreated to the line.
As the visitors wobbled, Gordon made a double save from Bertrand Traore but the Tynecastle side finished the first half holding firm.
The Turkish side started the second half strongly, though, 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 from a Gordon throw-out, Omer Ali Sahiner squared for Ozcan, just on the pitch, and he sorted out his feet out before hammering the ball high into the net.
Hearts kept going in an attempt to grab a goal to give their travelling fans something to cheer and they got that in the 90th minute when keeper Muhammed Sengezer parried a shot from Shankland with Atkinson on hand to knock the ball into the net.