The Germany international guided home Jason Denayer's flick-on after half an hour to settle a final high on energy but low on quality, with Nani sent off following the final whistle after appearing to sarcastically applaud the referee.
The result means Galatasaray, who finished sixth in the Super Lig, will serve their one-year ban from European football – imposed by UEFA for not complying with Financial Fair Play rules – next season, making them eligible once more in the 2017-18 campaign.
Podolski's goal would have come as a relief for the former Arsenal striker as it allowed him to atone for a glaring fifth-minue miss. Emre Colak whipped in a free-kick that was diverted Podolski's way, but he could only fire his half-volley tamely over at the far post despite being unmarked five yards from goal.
The 30-year-old did better seven minutes later with a blistering drive following a clever set-piece routine that drew a smart save from Fener goalkeeper Fabiano.
Denayer deflected Alper Potuk's shot wide at the other end before Podolski did find the back of the net. Emre's inswinging corner was helped on by on-loan Manchester City defender Denayer and Podolski diverted it past Fabiano.
Fener gradually grew into the game although chances remained at a premium, captain Mehmet Topal coming closest with a free header that was straight at Fernando Muslera.
Netherlands internationals Robin van Persie and Wesley Sneijder were withdrawn at the break and the game suffered as a result, with the first 15 minutes of the second period devoid of noteworthy incidents.
The tedium was finally broken in the 62nd minute as Muslera flapped at a Nani free-kick and Abdoulaye Ba nodded against the angle of post and crossbar.
Gala responded almost immediately as Fabiano saved well to keep out a Denayer header.
Fener ramped up the pressure in the final quarter of the game, but substitute Fernandao wasted their best chance with a wayward header after a fine cross from Nani.
Aurelien Chedjou almost put through his own net in injury-time while Simon Kjaer thrashed a last-gasp volley high over the bar, and Fener's woes were compounded by Nani's post-whistle dismissal.