Lukas Podolski got the Gunners, who finished second in Group D despite the big win, off to a flying start in Istanbul with a thumping strike in the third minute, before Ramsey doubled his side???s advantage just eight minutes later.
The hosts had been blown away by Arsene Wenger???s side???s clinical performance and they were left shell-shocked when Ramsey produced a stunning volley from 30-yards to give his side a 3-0 lead going into the break.
Wesley Sneijder pulled one back for the hosts late in the game but it was Arsenal who had the final say as Podolski picked up his second to seal a comprehensive European win.
However, the result was not enough to secure top spot in the group. Borussia Dortmund’s 1-1 draw with Anderlecht means they must anxiously wait for next Monday???s second-round draw where European heavyweights Real Madrid and Bayern Munich lie in wait as possible opponents.
Wenger took the decision to leave several key players at home for the trip to Istanbul, but despite the absence of Alexis Sanchez, Olivier Giroud, Santi Cazorla, Danny Welbeck and Laurent Koscielny the Gunners put their nightmare at Stoke behind them with a scintillating display in Turkey.
Podolski got the visitors off to a perfect start after brilliant work from Alex Oxlade-Chamberlain and Ramsey released the German into space. Podolski did not need a second invitation to get a shot away and his strike gave keeper Sinan Bolat no chance, nestling in the roof of the net.
The home side responded well and had an excellent chance to equalise when the ball fell at the feet of Hakan Balta, but the centre-back screwed his shot wide from close range.
The miss was to prove costly as Ramsey, fed by the impressive Oxlade-Chamberlain, kept his composure in front of goal before side-footing into the far corner to double his side???s advantage in the 10th minute.
The home fans were in a state of shock and things could have got even worse in the 20th minute as Podolski found space on the on the edge of the area, but his left-footed shot rattled the crossbar.
However, it was not long before Arsenal were on the scoresheet again as Ramsey, who has now scored six goals in six European appearances against Turkish opposition, produced one of the best goals of the season so far.
Joel Campbell???s corner was cleared at the near post, but what followed next was pure brilliance as the Welshman???s left-footed volley arrowed straight into the top corner to give the Gunners a 3-0 lead at half-time.
With Ramsey complaining of a tight hamstring during the interval and Mathieu Flamini already on a yellow card, Wenger took no chances with Gedion Zelalem and Ainsley Maitland-Niles introduced after the break. The changes stemmed the flow of Arsenal???s attacking play and it was Galatasaray who looked the most likely for much of the second half.
Substitute Hamit Altintop made an immediate impact, turning inside a tired Mathieu Debuchy but his shot was safely gathered by Wojciech??Szczesny.
In the 66th minute, it took a last-ditch block from Per Mertesacker, who received plenty of criticism for his performance in Arsenal???s 3-2 defeat at Stoke last weekend, to keep Umut Bulut at bay from point blank range.
Galatasary eventually got some reward for the efforts with just three minutes remaining as Sneijder produced a moment of magic to beat Szczesny with a curling free-kick to score the Turkish side’s 100th Champions League goal.
But on a night when Arsenal got back to winning ways it was Podolski who had the final word as he latched onto Yaya Sanogo???s ball to slot home a controlled finish from the centre of the penalty area.
Paul Merson’s verdict
???Ramsey is the type of lad that can go and score six in six, without a doubt. It???s all about confidence and we???ve seen that he can do it ??? he???s a really, really good player. If he had reached the heights of last season he???d be playing for Barcelona. That???s how high and how good he was. I thought he was the best midfield player in the country last year ??? probably the best player in the country at one stage, with Luis Suarez, until he got injured.”
Galatasaray: Sinan Bolat (5), Tarik Camdal (5), Semih (5), Hakan (6), Telles (4), Melo (5), Emre Colak, Bruma (6), Sneijder (7), Umut Bulut (6), Burak Yilmaz (5).
Used subs: Altintop (7), Oztekin (6), Adin (5).
Arsenal: Szczesny (7), Debuchy (7), Chambers (6), Mertesacker (7), Bellerin (7), Ramsey (9), Flamini (6), Oxlade-Chamberlain (8), Campbell (6), Sanogo (6), Podolski (8).
Used subs: Zelalem (5), Maitland-Niles (5), O???Connor (5).
Man of The Match: Aaron Ramsey