Pierre Webo's 79th-minute penalty and Dirk Kuyt's last-minute finish ensured the hosts eventually made their second-half pressure pay.
Ogenyi Onazi's red card three minutes after the break handed the initiative to the Turks, but, after hitting a post and seeing a succession of other chances come and go, they looked in danger of being frustrated.
But January signing Webo held his nerve after Stefan Radu had handled in the area to opening the scoring and Kuyt added a second on the rebound at the death.
It all leaves Lazio with plenty of work to do in the second leg in Rome.
Fener's Moussa Sow had the first chance for the hosts after four minutes, but his left-footed attempt from the left side of the area was pulled wide of the far post.
At the other end Ederson Honorato met Antonio Candreva's cross, but his header from six yards was kept out by Volkan Demirel.
Sow, who has been a hit in Istanbul since his signing from Lille in January last year, went closest to breaking the deadlock in the 17th minute when his right-footed attempt from the centre of the area came back off a post.
Neither side were able to get a grip on the game or carve out a single chance of note as the half stagnated, but the visitors shot themselves in the foot three minutes after the restart.
Onazi had already been booked in the first half and so could have no complaints when he picked up a second yellow for scything down Cristian Baroni.
The hosts looked to take immediate advantage, but Raul Meireles' shot from the edge of the area struck a post.
Sow then fired wide from much closer range before Meireles had a shot tipped round a post by Lazio keeper Federico Marchetti.
Fener kept up the pressure, Webo heading a Kuyt cross wide and Sow forcing another smart save from Marchetti, until the breakthrough finally arrived.
Radu was penalised for a handball following Caner Erkin's corner and Webo stepped up to slot the resulting spot-kick low into the left corner.
Lazio would have been happy to leave Turkey with just a one-goal deficit, but their opponents were keen to press home their advantage and a second arrived in the 90th minute.
Erkin's free-kick was parried by Marchetti and Kuyt followed in to score.