The £16 million man sped past a shoddy Turkish defence to prod home three clever goals as the Gunners got their group-stage campaign off and running despite losing their keeper
The England international struck twice in the first half and once more in the second, frequently darting past a hapless back line with clever poked finishes each time.
Alexis Sanchez hit Arsenal's third just before the break, slotting home with precision into the far corner while under pressure in the penalty area, as Arsene Wenger's side registered their first points of the group stage.
It was not a night without problems for the Gunners, though, as midway through the second half Szczesny darted off his line in a vain attempt to snatch the ball from the feet of Burak Yilmaz, succeeding only in fouling the striker, who then put the resulting penalty past substitute David Ospina.
The visitors perked up a little as they tested Ospina but Arsenal could have had more, Santi Cazorla being denied as his prodded finish was narrowly cleared off the line in the dying moments.
Mathieu Flamini, Cazorla and Alexis came into the Arsenal XI as a result of midfield trio Mikel Arteta, Aaron Ramsey and Jack Wilshere sustaining injuries during Saturday's 1-1 draw against Tottenham. The latter was able to take a place on the bench despite his ankle complaint.
Galatasaray captain Selcuk Inan was a late withdrawal due to illness, meaning that Yekta Kurtulus came into midfield as Cesare Prandelli plumped for a 3-5-2 formation to which his side struggled to adapt during a challenging opening period.
Visiting defender Felipe Melo made a vital interception to stop Welbeck getting on the end of Alexis's low 20th-minute cross but the same combination put Arsenal ahead two minutes later.
The Chilean cut in from the left flank and slid a pass through to Welbeck, who peeled off the back of Melo to slot home.
Galatasaray made matters worse in the 30th minute as an ill-judged back header from Aurelien Chedjou sold Melo short and Welbeck left the Brazilian in his wake to finish clinically.
A torrid evening for Melo might have ended early when he was only booked for a two-footed lunge on Alexis but, after a section of the Galatasaray support interrupted proceedings by throwing flares onto the pitch, it was the Chilean winger who ran riot - collecting Mesut Ozil's pass and scorching through the visitors' beleaguered defence to find the bottom corner.
Kurtulus made way for Hamit Altintop at half-time as Prandelli reverted to 4-4-2 but it did little to stem the tide.
Alex Oxlade-Chamberlain became the latest Arsenal player to torment Melo, forcing Dzemaili to clear from the goalmouth, and the England midfielder was the architect when Welbeck completed his treble. A beautifully judged pass through Alex Telles's legs was collected by Oxlade-Chamberlain's England team-mate, who dinked nonchalantly beyond Muslera.
Pandev's slide-rule pass caught the Arsenal back four flat-footed after an hour and Szczesny darted out to be sent off as he was in last season's second-round loss to Bayern Munich at the Emirates Stadium.
Burak made no mistake from the spot, sending David Ospina the wrong way, although the replacement goalkeeper saved impressively from the goalscorer and Altintop in the final 10 minutes before Cazorla saw an effort chacked off the line by Semih Kaya at the other end.
Galatasaray now have it all to do in Group D - this thrashing following their opening 1-1 draw at home to Anderlecht.