Podolski got the Gunners off to a flyer, firing home after just three minutes to put Arsene Wenger's men ahead in Istanbul.
Ramsey then doubled his side's advantage seven minutes later, running on to a great ball from Alex Oxlade-Chamberlain and aiming past the keeper.
Arsenal v Newcastle - Premier League Preview
Ramsey grabbed his second of the night just before the half-hour mark, unleashing an unstoppable 35-yard drive to put his team firmly in the driving seat.
Wenger's team controlled much of the play for the second half, but former Real Madrid and Inter Milan maestro Wesley Sneijder pulled one back for the Turks with just minutes remaining.
However, Podolski made sure there were going to be no upsets, burying the ball in the corner with just seconds remaining and sealing Arsenal's second-place position behind group winners Borussia Dortmund.
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