Following a goalless first leg, Jack Cork scored the only goal at Turf Moor when he curled into the top corner.

Earlier, Ashley Barnes shot wide and Phil Bardsley's strike was saved by goalkeeper Mert Gunok.

Burnley goalkeeper Joe Hart made fine saves to deny Edin Visca and Kerim Frei at the end of the first half.

The Clarets, in their first season of European football since 1966-67, will play Greek side Olympiakos in a two-legged play-off, with the winners qualifying for the group stage.

"The players are showing a desire for the competition, which is great because there is a lot of negative noise around it," manager Sean Dyche told Radio 5 live.

"It is a massive badge of honour to be in a European competition, as most fans thought those days had gone forever."

Chelsea and Arsenal will join the competition in the group phase.

Another clean sheet for Hart

Dyche's Burnley, who qualified for Europe by finishing seventh in the Premier League, have already played four matches in the competition.

They needed extra time to overcome Aberdeen 4-2 on aggregate in the second qualifying round this month, and again had to battle hard to edge out Istanbul Basaksehir.

Burnley managed only two shots in the first leg - their lowest in any match since March 2012 - with neither on target.

But they made a brighter start on Thursday as Barnes hit the side-netting and Bardsley's powerful strike was well saved.

There were no away fans in the ground after Istanbul Basaksehir made the decision not to sell tickets to their fans, but the visitors enjoyed a bright spell just before half-time.

Joe Hart has not conceded a goal in the three games he has played for Burnley since moving from Manchester City earlier this month

This was a third consecutive clean sheet for Burnley since England international Hart joined for £3.5m from Manchester City this month.

The 31-year-old, left out of England's squad for this summer's World Cup, twice denied Visca and also frustrated former Fulham midfielder Frei to help take the tie into extra time.

Despite his heroics, Dyche was keen for Hart not to become the focal point of the Clarets' season.

"I'm surprised you're not fed up, we've been talking about him for two weeks," said Dyche.

"I made a lot of the display of the team. There were a lot of really good performances and they deserve just as many mentions.

"But Joe definitely played his part, just so you're happy."

After the first leg, Istanbul Basaksehir manager Abdullah Avci criticised Burnley's "long-ball" tactics, but Cork's 98th-minute goal came via a fine finish from 22 yards as he curled home after receiving a pass from Jeff Hendrick.

"Tonight's was a sublime goal," added Dyche. "That is something we don't get enough credit for. If that is a more trendy name than Jack Cork, it would be raved about."

Former Arsenal, Manchester City and Tottenham striker Emmanuel Adebayor came on in the 66th minute for the visitors, but shot over in normal time and headed over late in extra time.

What next for Burnley?

Burnley face Olympiakos in the Europa League play-off round, with the first leg in Greece taking place next Thursday and the second leg at Turf Moor on 30 August.

Olympiakos, who finished third in the Greek Superleague last season, recorded a 7-1 aggregate win over Swiss side Luzern in the third qualifying round - a 4-0 victory in Greece in the first leg followed by a 3-1 victory in Switzerland on Thursday.

The Olympiakos squad includes former Nottingham Forest midfielder Andreas Bouchalakis, defender Omar Elabdellaoui, who began his career at Manchester City and had a loan spell at Hull last season, and centre-half Roderick Miranda, on loan from Wolves.

Burnley have a home Premier League game against Watford on Sunday (13:30 BST).