The Gunners dominated for long spells and seized control when Wilshere collected a clever pass from Nacho Monreal and drilled a low shot into the corner on 34 minutes.
The England midfielder then turned provider just seconds later when he took advantage of an error from Fabian Delph to feed Giroud, who made no mistake with a clinical finish across Brad Guzan.
Villa had suffered an early blow when Nathan Baker was forced off with mild concussion, but they rallied after the break and reduced the arrears when Christian Benteke headed home Matt Lowton's probing cross.
The Belgian's goal ended an 11-match drought and, clearly revitalised by the goal, he almost saved the day late on but saw his header saved.
The result sees Arsenal move one point clear of Manchester City on a night that also saw Alex Oxlade-Chamberlain make a late cameo appearance after a lengthy spell on the sidelines.
PRE-MATCH: Villa switch formation, the Ox returns - It's a 3-5-2 formation for Villa with captain Ron Vlaar returning to shore up the defence. Christian Benteke and Gabriel Agbonlahor lead the attack. For Arsenal, Olivier Giroud ploughs a lone furrow up top and Serge Gnabry comes into the side. Alex Oxlade-Chamberlain is back in the fold for the first time since the opening day defeat to Villa, and is named on the bench. Mikel Arteta misses out with a calf problem.
5' - ARSENAL CHANCE. Bacary Sagna bends a lovely cross into the middle for Giroud who guides his free header way off target. Big miss from the Frenchman. You'd expect him to gobble those type of chances up.
21' INJURY BLOW FOR VILLA. A warm ovation for Nathan Baker who is carried off on a stretcher after a six-minute delay. The defender is suffering form mild concussion.
23' VILLA CHANCE. El Ahmadi side foots a volley just wide after Delph had teased in a cross from the left. Nice link up by Agbonlahor and Benteke in the lead up to that counter attack.
34' - GOAL: Aston Villa 0-1 Arsenal. Monreal breaks free down the left and tucks the ball back for Wilshere who takes a decent first touch before firing low into the far corner.
35' - GOAL: Villa 0-2 Arsenal. It's two in a minute for the Gunners as Giroud capitalises upon Wilshere's clipped pass and drills a clinical shot beyond Guzan. Poor play by Delph who gave away the ball cheaply in his own half.
76' - GOAL! Villa 1-2 Arsenal. At last. Benteke dives in at the far post to head home Lowton's delightful right-wing cross to end his barren spell in front of goal.
90' VILLA CHANCE! A cross from the right reaches Benteke who guides his header straight at the keeper.
The Gunners’ 15 game unbeaten run at Villa is their longest away run without defeat against a single side in Premier League history.
Villa have kept just two clean sheets in their last 22 home league matches.
The 59 second gap between Arsenal's first and second goals versus Villa is the shortest seen between two goals in a PL game this season. Flash.
No team has scored the first goal in more Premier League games this season than Arsenal (16). Conversely, no side has scored the first goal on fewer occasions this season than Aston Villa (5).
Aston Villa striker Christian Benteke scored his first Premier League goal for 14 hours and 45 minutes.
(Stats courtesy of Opta)
MAN OF THE MATCH
Jack Wilshere. (Arsenal): A goal and an assist from the midfielder who relished the chance to play in a more advanced role. This form is good news for Arsenal and england.
PLAYER RATINGS -
ASTON VILLA: Guzan 6, Lowton 6, Vlaar 6, Clark 6, Baker 5, Luna 5, El Ahmadi 6, Westwood 6, Delph 5, Agbonlahor 6, Benteke 7. Subs: Bacuna 5, Weimann 6.
ARSENAL: Szczesny 6, Sagna 7, Mertesacker 7, Koscielny 7, Monreal 7, Wilshere 8, Flamini 7, Cazorla 7, Ozil 5, Gnabry 6, Giroud 7. subs: Gibbs 6, Rosicky 6, Oxlade-Chamberlain 5.
THE MATCH IN A TWEET