The Brazilian, a summer signing from Manchester City, scored two and set up the other for Milan Baros as the Turkish giants took control of the game.
Dimitris Salpingidis got a late goal back for the hosts but only after the match was beyond their reach.
Galatasaray opened the scoring with the first clear chance of the game as Baros' cross picked out Elano and the Brazilian rifled in his second goal for the club from close range.
That stirred Pana into action, but Argentinian goalkeeper Leo Franco dealt with Sebastian Leto's free-kick comfortably as the hosts got their first sight of goal.
Elano went close to a second at the other end but Mario Galinovic was able to keep out a set-piece from the midfielder.
Nikos Spiropoulos blazed over as the hosts would regret their profligacy as the chances dried up at both ends.
The match did not come back to life until the final five minutes before the break, when Harry Kewell tested Galinovic from the edge of the area.
At the other end, Giorgios Karagounis sent a shot wide before Kostas Katsouranis missed the target with a long-range effort.
If the home fans thought those were signs of a revival, their hopes were dashed almost immediately after the restart.
In a reverse of roles from the first goal, Elano crossed from the right and Baros slid the ball into the bottom left corner of the goal.
Pana scrambled for a response, but Leto missed the target before Franco kept out Sotiris Ninis' shot.
Antonis Petropoulos then missed from close range and moments later Pana's hopes were ended as Elano supplied the killer blow, firing in a free-kick after Spiropoulos brought down Mustafa Sarp 25 yards from goal.
His work done for the night, Elano was soon withdrawn as Galatasaray were happy to let Pana come forward in search of a goal that seemingly would not come.
Ninis and Petropoulos carved out a couple of chances each but neither could find the target.
The breakthrough finally came in the 78th minute as Karagounis crossed for Salpingidis to finish from close range.
Salpingidis missed another chance moments later and, having received the wake-up call, Galatasaray tightened things up to see the game out and collect the three points.