The Clarets needed extra time to continue their first European campaign since 1966-67, but Jack Cork's 98th-minute strike sealed their place in the next round after the first leg in Turkey ended goalless last week.
Burnley will now face Greek side Olympiacos in the playoffs, with a place in the group stages of the competition awaiting the winners.
The hosts failed to have a single shot on target and managed just two attempts overall in the first leg last week, but it took only 10 minutes for them to carve out their first chance of the game this evening when Ashley Barnes broke forward and rippled the side-netting with his strike.
Just five minutes later Burnley had already equalled their shot count from Turkey - and bettered their tally on target - courtesy of Phil Bardsley's thunderbolt from range, which required Mert Gunok to pull off a fine stop to prevent it from flying into the top corner.
Sean Dyche's side continued to look the more likely as the first half wore on and Barnes had another sight of goal midway through the opening period, although this time his volley on the turn sailed well over the crossbar.
The hosts did not look like a team that had recorded back-to-back goalless draws prior to this match, but the opening goal continued to elude them when Aaron Lennon danced into the box before drawing a good stop from Gunok, who atoned for his own error having initially given the ball away cheaply.
It wasn't until the 42nd minute that Joe Hart - starting for the third match in a row - was called into action for the first time, but the former England number one still needed to be alert to palm Edin Visca's effort away.
It proved to be a busy end to the half for Hart, though, and he needed to be at his best to deny Visca again moments later when the Bosnian latched on to Kerim Frei's pass before going for goal from close range.
A third stop followed in first-half stoppage time as Basaksehir enjoyed a late flurry of chances, this time keeping out Frei's curling effort on the stroke of the interval.
The second half was not quite as action-packed in the final third as the tension cranked up for both sides, although Burnley did come close shortly after the hour mark when Sam Vokes saw his header saved by Gunok.
The tie looked increasingly set for extra time, though, and sure enough it went to an additional 30 minutes despite Basaksehir providing their hosts with a late scare through Mahmut Tekdemir's effort, which fizzed narrowly wide two minutes from time.
Burnley finally made the breakthrough midway through the first half of extra time, and it was a goal worth waiting for as Cork curled his finish into the top corner from outside the area to put his side on the verge of the playoffs.
There was still work to be done for the Clarets, though, and a familiar face almost brought them back down to earth when former Arsenal, Tottenham Hotspur and Manchester City striker Emmanuel Adebayor nodded just over from close range five minutes later.
That was as close as the visitors came to forcing penalties, though, as Burnley held out for the victory which takes them one step closer to the group stages of the competition.
BURNLEY (5-3-1-1): Hart; Bardsley, Ward, Long, Gibson, Taylor; Lennon (Gudmundsson (58'), Westwood (Cork 82'), Hendrick; Barnes (Tarkowski 122'); Vokes (Wood 82')
ISTANBUL BASAKSEHIR (4-2-3-1): Gunok; Da Costa, Clichy, Epureanu, Caicara; Emre, Tekdemir (Inler 92'); Visca, Frei (Napoleoni (102'), Kahveci (Jojic 109'); Bajic (Adebayor 66')