Arsenal were bidding to bounce back from their shock weekend defeat to West Brom but the north Londoners looked rattled in the early stages in Serbia and could easily have fallen behind.
However, Arshavin fired the Gunners ahead from 12 yards in the 15th minute and saw an effort cleared off the line not long after as the visitors began to dominate.
Cleo drew Partizan level from the penalty spot in the 33rd minute after Denilson handballed in the box and Arsenal missed the chance to go back ahead 10 minutes into the second half when Arshavin missed a spotkick after Marko Jovanovic saw red for hauling down Chamakh.
However, the Gunners did go back ahead in the 71st minute when Chamakh nodded in a rebound after his initial header was saved.
Squillaci headed in a third from Sami Nasri's corner eight minutes from time and Lukasz Fabianski ensured there would be no late drama with a fine save from another Cleo spot-kick, this time given for a foul by Kieron Gibbs.
The Pole capped a confident display with another acrobatic stop from Ivica Iliev in stoppage time as Arsenal recorded their first European away win in a year.
There was drama before the game when the ground was plunged into darkness due to a power failure, with the players forced to change in the dark.
Power was restored, and then failed in the main stand again just as the teams walked out to a cacophony of noise at Stadion FK Partizan, which was transformed into a sea of black and white as the 'Gravediggers' threw paper and waved flags, unveiling a banner declaring: "You have been run over by a steamroller."
The Serbians have a chequered history of crowd trouble, thrown out of the 2007-08 UEFA Cup while in September last year, a Toulouse fan died in hospital after being injured following violence in a bar ahead of the two sides' Europa League tie.
Indeed, Partizan went as far as issuing a plea for calm on their own website ahead of tonight's game.
Wenger, though, maintained his men should be able to cope with such an intimidating atmosphere, and so it proved.
The Gunners, playing in yellow, made a positive start, as Jack Wilshere was pressed up in support of lone frontman Chamakh.
The home fans, though, were screaming for a penalty when Pierre Boya went down under close attention from Johan Djourou, but German referee Wolfgang Stark was having none of it.
Cleo then tripped over as he got clear into the left side of the Arsenal box, with Denilson sliding in to make a saving tackle as Nemanja Tomic looked to capitalise on the loose ball.
Fabianski needed two attempts to gather a hesitant header back to him from Squillaci as Boya looked to nip in on goal.
Cleo was lively, dispossessing Djourou on the right before cutting in to blast over from 20 yards.
However, the home side were stunned on 15 minutes as Arsenal snatched the lead.
Wilshere charged at the defence and when his path was blocked, the England youngster had the presence of mind to back-heel the ball into the path of Arshavin, who drilled his shot into the bottom corner.
The Russian almost grabbed a second when put into the penalty area, but Partizan goalkeeper Vladimir Stojkovic was out quickly.
Stojkovic denied Tomas Rosicky, before Jovanovic cleared Arshavin's goal-bound chip.
Arsenal paid the penalty for missing so many chances though when Denilson needlessly stuck his arm out as Radosav Petrovic delivered the ball into the box after half an hour.
Cleo sent Fabianski - in for the injured Manuel Almunia - the wrong way to level.
Arsenal had an early chance at the start of the second half following a driving run by Alex Song, with the ball breaking to Rosicky on the right and his angled drive was well held by the goalkeeper.
Partizan were reduced to 10 men in the 56th minute when Jovanovic was shown a red card for tripping Chamakh as the Moroccan latched onto Arshavin's through ball.
Arshavin took the penalty himself - only to blast it straight at the goalkeeper.
On 69 minutes, the lively Arshavin was again denied, this time when Stojkovic made a good stop at the near post.
Moments later, though, and it was 2-1.
Rosicky floated the ball in from the right and Chamakh headed goalwards. Stojkovic touched the ball onto the crossbar but the Arsenal striker was first to the rebound, nodding the ball into an empty net.
With eight minutes left, Squillaci headed in a corner from substitute Samir Nasri to make it 3-1.
There was late drama, though, when Kieran Gibbs was adjudged to have tripped Ivan Stevanovic in the penalty area, but this time Fabianski produced a brilliant save to deny Cleo before another fine save to stop Iliev.